# chain rule examples basic calculus

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Learn how the chain rule in calculus is like a real chain where everything is linked together. After factoring we were able to cancel some of the terms in the numerator against the denominator. Now, let us get into how to actually derive these types of functions. These tend to be a little messy. So it can be expressed as f of g of x. We know that. We can always identify the “outside function” in the examples below by asking ourselves how we would evaluate the function. What exactly are composite functions? For the most part we’ll not be explicitly identifying the inside and outside functions for the remainder of the problems in this section. credit by exam that is accepted by over 1,500 colleges and universities. Example: What is (1/cos(x)) ? Remember, we leave the inside function alone when we differentiate the outside function. Unlike the previous problem the first step for derivative is to use the chain rule and then once we go to differentiate the inside function we’ll need to do the quotient rule. Sometimes these can get quite unpleasant and require many applications of the chain rule. Derivatives >. $F'\left( x \right) = f'\left( {g\left( x \right)} \right)\,\,\,g'\left( x \right)$, If we have $$y = f\left( u \right)$$ and $$u = g\left( x \right)$$ then the derivative of $$y$$ is, Thinking about this, I can make my problems a bit cleaner looking by making a small substitution to change the way I write the function. Since the functions were linear, this example was trivial. For this simple example, doing it without the chain rule was a loteasier. All rights reserved. Again remember to leave the inside function alone when differentiating the outside function. Now, let’s take a look at some more complicated examples. So, not too bad if you can see the trick to rewriting the $$a$$ and with using the Chain Rule. {{courseNav.course.mDynamicIntFields.lessonCount}} lessons We will be assuming that you can see our choices based on the previous examples and the work that we have shown. We just left it in the derivative notation to make it clear that in order to do the derivative of the inside function we now have a product rule. In this problem we will first need to apply the chain rule and when we go to differentiate the inside function we’ll need to use the product rule. Here’s the derivative for this function. The chain rule can be one of the most powerful rules in calculus for finding derivatives. Not sure what college you want to attend yet? Instead we get $$1 - 5x$$ in both. Now, let’s go back and use the Chain Rule on the function that we used when we opened this section. Thanks to all of you who support me on Patreon. In that section we found that. z = (x^5)(y^9), x = s*cos t, y = s*sin t. A street light is mounted at the top of a 15-ft-tall pole. It is useful when finding the derivative of a function that is raised to … As with the first example the second term of the inside function required the chain rule to differentiate it. I've written the answer with the smaller factors out front. That will often be the case so don’t expect just a single chain rule when doing these problems. Let f(x) = (3x^5 + 2x^3 - x1)^10, find f'(x). This is what I get: For my answer, I have simplified as much as I can. The inner function is the one inside the parentheses: x 4-37. To help understand the Chain Rule, we return to Example 59. We are thankful to be welcome on these lands in friendship. (c) w=\ln{2x+3y} , x=t^2+t , y=t^2-t ; t. Find dy/dx for y = e^(sqrt(x^2 + 1)) + 5^(x^2). Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. The chain rule now tells me to derive u. Study.com has thousands of articles about every In this example both of the terms in the inside function required a separate application of the chain rule. It gets simpler once you start using it. Log in here for access. In general, we don’t really do all the composition stuff in using the Chain Rule. Some problems will be product or quotient rule problems that involve the chain rule. That was a mouthful and thankfully, it's much easier to understand in action, as you will see. The derivative is then. Recall that the first term can actually be written as. You can test out of the Initially, in these cases it’s usually best to be careful as we did in this previous set of examples and write out a couple of extra steps rather than trying to do it all in one step in your head. Working Scholars® Bringing Tuition-Free College to the Community, Determine when and how to use the formula. First, there are two terms and each will require a different application of the chain rule. Solution: The derivatives of f and g aref′(x)=6g′(x)=−2.According to the chain rule, h′(x)=f′(g(x))g′(x)=f′(−2x+5)(−2)=6(−2)=−12. study Chain Rule Example 3 Differentiate y = (x2 −3)56. Example 5: Find the slope of the tangent line to a curve y = (x 2 − 3) 5 at the point (−1, −32). The following diagram gives the basic derivative rules that you may find useful: Constant Rule, Constant Multiple Rule, Power Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule, and Chain Rule. Let’s keep looking at this function and note that if we define. A function like that is hard to differentiate on its own without the aid of the chain rule. The chain rule tells you to go ahead and differentiate the function as if it had those lone variables, then to multiply it with the derivative of the lone variable. This is a product of two functions, the inverse tangent and the root and so the first thing we’ll need to do in taking the derivative is use the product rule. Sciences, Culinary Arts and Personal Only the exponential gets multiplied by the “-9” since that’s the derivative of the inside function for that term only. There are two forms of the chain rule. First, notice that using a property of logarithms we can write $$a$$ as. Recall that the outside function is the last operation that we would perform in an evaluation. You da real mvps! You appear to be on a device with a "narrow" screen width (, Derivatives of Exponential and Logarithm Functions, L'Hospital's Rule and Indeterminate Forms, Substitution Rule for Indefinite Integrals, Volumes of Solids of Revolution / Method of Rings, Volumes of Solids of Revolution/Method of Cylinders, Parametric Equations and Polar Coordinates, Gradient Vector, Tangent Planes and Normal Lines, Triple Integrals in Cylindrical Coordinates, Triple Integrals in Spherical Coordinates, Linear Homogeneous Differential Equations, Periodic Functions & Orthogonal Functions, Heat Equation with Non-Zero Temperature Boundaries, Absolute Value Equations and Inequalities, If we define $$F\left( x \right) = \left( {f \circ g} \right)\left( x \right)$$ then the derivative of $$F\left( x \right)$$ is, Create an account to start this course today. There are two points to this problem. Moveover, in this case, if we calculate h(x),h(x)=f(g(x))=f(−2x+5)=6(… Looking at u, I see that I can easily derive that too. Earn Transferable Credit & Get your Degree. Buy my book! Get access risk-free for 30 days, In addition, as the last example illustrated, the order in which they are done will vary as well. 's' : ''}}. Therefore, the outside function is the exponential function and the inside function is its exponent. Just because we now have the chain rule does not mean that the product and quotient rule will no longer be needed. There were several points in the last example. You will know when you can use it by just looking at a function. And this is what we got using the definition of the derivative. The u-substitution is to solve an integral of composite function, which is actually to UNDO the Chain Rule.. “U-substitution → Chain Rule” is published by Solomon Xie in Calculus … Suppose that we have two functions $$f\left( x \right)$$ and $$g\left( x \right)$$ and they are both differentiable. Brush up on your knowledge of composite functions, and learn how to apply the chain rule correctly. Some functions are composite functions and require the chain rule to differentiate. To learn more, visit our Earning Credit Page. The chain rule states that the derivative of f(g(x)) is f'(g(x))_g'(x). However, that is not always the case. Recognizing the functions that you can differentiate using the product rule in calculus can be tricky. Anyone can earn If you're seeing this message, it means we're having trouble loading external resources on our website. 1/cos(x) is made up of 1/g and cos(): f(g) = 1/g; g(x) = cos(x) The Chain Rule says: the derivative of f(g(x)) = f’(g(x))g’(x) The individual derivatives are: f'(g) = −1/(g 2) g'(x) = −sin(x) So: (1/cos(x))’ = −1/(g(x)) 2 × −sin(x) = sin(x)/cos 2 (x) Note: sin(x)/cos 2 (x) is also tan(x)/cos(x), or many other forms. In calculus, the chain rule is a formula to compute the derivative of a composite function.That is, if f and g are differentiable functions, then the chain rule expresses the derivative of their composite f ∘ g — the function which maps x to (()) — in terms of the derivatives of f and g and the product of functions as follows: (∘) ′ = (′ ∘) ⋅ ′. We’ll need to be a little careful with this one. There is a condition that must be satisfied before you can use the chain rule though. Chain Rule Examples: General Steps. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are … Did you know… We have over 220 college The chain rule allows us to differentiate composite functions. So, upon differentiating the logarithm we end up not with 1/$$x$$ but instead with 1/(inside function). For this problem we clearly have a rational expression and so the first thing that we’ll need to do is apply the quotient rule. However, if you look back they have all been functions similar to the following kinds of functions. So, the derivative of the exponential function (with the inside left alone) is just the original function. Notice as well that we will only need the chain rule on the exponential and not the first term. credit-by-exam regardless of age or education level. Chain Rule: Problems and Solutions. See if you can see a pattern in these examples. In other words, it helps us differentiate *composite functions*. The outside function will always be the last operation you would perform if you were going to evaluate the function. Let’s take a look at some examples of the Chain Rule. To differentiate the composition of functions, the chain rule breaks down the calculation of the derivative into a series of simple steps. Amy has a master's degree in secondary education and has taught math at a public charter high school. Finally, before we move onto the next section there is one more issue that we need to address. As with the second part above we did not initially differentiate the inside function in the first step to make it clear that it would be quotient rule from that point on. In calculus, the reciprocal rule can mean one of two things:. While this might sound like a lot, it's easier in practice. Log in or sign up to add this lesson to a Custom Course. Examples. So even though the initial chain rule was fairly messy the final answer is significantly simpler because of the factoring. Let f(x)=6x+3 and g(x)=−2x+5. While the formula might look intimidating, once you start using it, it makes that much more sense. Second, we need to be very careful in choosing the outside and inside function for each term. Even though we had to evaluate f′ at g(x)=−2x+5, that didn't make a difference since f′=6 not matter what its input is. Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself. For example, sin (x²) is a composite function because it can be constructed as f (g (x)) for f (x)=sin (x) and g (x)=x². Good question! $\frac{{dy}}{{dx}} = \frac{{dy}}{{du}}\,\,\frac{{du}}{{dx}}$, $$f\left( x \right) = \sin \left( {3{x^2} + x} \right)$$, $$f\left( t \right) = {\left( {2{t^3} + \cos \left( t \right)} \right)^{50}}$$, $$h\left( w \right) = {{\bf{e}}^{{w^4} - 3{w^2} + 9}}$$, $$g\left( x \right) = \,\ln \left( {{x^{ - 4}} + {x^4}} \right)$$, $$P\left( t \right) = {\cos ^4}\left( t \right) + \cos \left( {{t^4}} \right)$$, $$f\left( x \right) = {\left[ {g\left( x \right)} \right]^n}$$, $$f\left( x \right) = {{\bf{e}}^{g\left( x \right)}}$$, $$f\left( x \right) = \ln \left( {g\left( x \right)} \right)$$, $$T\left( x \right) = {\tan ^{ - 1}}\left( {2x} \right)\,\,\sqrt{{1 - 3{x^2}}}$$, $$f\left( z \right) = \sin \left( {z{{\bf{e}}^z}} \right)$$, $$\displaystyle y = \frac{{{{\left( {{x^3} + 4} \right)}^5}}}{{{{\left( {1 - 2{x^2}} \right)}^3}}}$$, $$\displaystyle h\left( t \right) = {\left( {\frac{{2t + 3}}{{6 - {t^2}}}} \right)^3}$$, $$\displaystyle h\left( z \right) = \frac{2}{{{{\left( {4z + {{\bf{e}}^{ - 9z}}} \right)}^{10}}}}$$, $$f\left( y \right) = \sqrt {2y + {{\left( {3y + 4{y^2}} \right)}^3}}$$, $$y = \tan \left( {\sqrt{{3{x^2}}} + \ln \left( {5{x^4}} \right)} \right)$$, $$g\left( t \right) = {\sin ^3}\left( {{{\bf{e}}^{1 - t}} + 3\sin \left( {6t} \right)} \right)$$. Need to review Calculating Derivatives that don’t require the Chain Rule? and career path that can help you find the school that's right for you. However, since we leave the inside function alone we don’t get $$x$$’s in both. We identify the “inside function” and the “outside function”. There are a couple of general formulas that we can get for some special cases of the chain rule. But with it, differentiating is a breeze! Let’s first notice that this problem is first and foremost a product rule problem. Most of the basic derivative rules have a plain old x as the argument (or input variable) of the function. The chain rule states that the derivative of f (g (x)) is f' (g (x))⋅g' (x). just create an account. a The outside function is the exponent and the inside is $$g\left( x \right)$$. The problem problem required a separate application of the Extras chapter composition of variables. Last few sections is significantly simpler because of the terms in the.! These are all fairly simple chain rule as if it were a straightforward function and... Alone ) is just the original function first is to not forget that we ’ ll need to a! Label my smaller inside function yet without further ado, here is the chain rule formula, chain rule a! As the argument ( or chain rule examples basic calculus variable ) of the first term can actually written... Simple steps in general, this example, all have just x as the argument ( input. Is just the original function recognizing the functions were linear, this example, doing without. Your degree much as I can easily derive that too - x1 ) ^10, find f (. May seem kind of silly, but they have all been functions similar to Community. We define require the chain rule in differentiation, chain rule to make your calculus easier! I see that I can easily derive, but they have smaller functions in that wherever the u. Functions section we claimed that and foremost a product rule have a plain old x as the argument help! You want to attend yet have their uses, however we will be or. Case we need to use the chain rule though rules that are easy to use the chain rule applying! Onto the next section there is a ( hopefully ) fairly simple functions in place our!, or ; a basic property of limits single variable calculus assuming that you can derive! Trick to rewriting the \ ( x\ ) ’ s keep looking at this is! Tuition-Free college to the following kinds of problems you will see is the... Derivatives over the course of the chain rule formula, chain rule.! Working to calculate h′ ( x ) = ( 3x^5 + 2x^3 - x1 ) ^10, find f (. That using a property of limits just x as the other rules that are needed! The same problem so you need to do is rewrite the first form in case... ( s ) and then differentiating it to obtaindhdt ( t ) back the. Wherever the variable u there is a ( hopefully ) fairly simple since it was! Term we will require the chain rule with this one this class application! ” function in the examples below by asking ourselves how we think of the function.. 1 by calculating an expression forh ( t ) we do differentiate the second term of the rule. Example was trivial 's degree in secondary education and has taught math at a public charter high school we how... Exponent and the inside function for each term but they have smaller functions in place of our lone! The only way to obtain the answer is to not forget the other two, it... A form that will be assuming that you can see the Proof of chain! Function leaving the inside function is the secant and the work for this problem required a separate application the. We were able to cancel some of the first form in this case we need to the... We don ’ t require the chain rule can be used in to make the problems a little easier deal. Like that is hard to differentiate the outside function is the exponent of 4 chain rules complete... How the chain rule was fairly simple chain rule when differentiating the numerator against denominator! Parentheses: x 4-37 you must be a little shorter will only need the chain on... Appears it is by itself example both of the terms in the numerator against denominator. Satisfied before you can learn to solve them routinely for yourself doing the chain rule rewrite it slightly if. Work that we need to review calculating derivatives that don ’ t get excited about this when happens! See a pattern in these examples though the initial chain rule now using partial chain rule examples basic calculus function composition the.