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The

\begin{align*}\Big[ f\Big(g(x)\Big)\Big]’ &= f’\Big(g(x)\Big) \cdot g'(x) \\[5px] &=\text{[derivative of the outer function, evaluated at the inner function] } \\[5px] &\qquad \times \text{ [derivative of the inner function]}

\end{align*}

In Leibniz notation:

\[\dfrac{dy}{dt} = \dfrac{dy}{du} \cdot \dfrac{du}{dt} \] And informally, the way you may quickly come to think about it:

\[\dfrac{df}{dx} = \left[\dfrac{df}{d\text{(stuff)}}\text{, with the same stuff inside} \right] \times \dfrac{d}{dx}\text{(stuff)}\]

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The goal of this page is for

And even though we’re saying these are “beginning problems,” you’ll find some toward the bottom that are from past exams at some of the world’s best-known science and engineering universities.

We’ll start by seeing how the Chain Rule works with the Power Rule, Exponentials, Trig Functions, and then the Product and Quotient Rules.

Chain Rule and Power Rule

\begin{align*}

\text{If} && f(x) &= (\text{stuff})^n, \\[8px]
\text{then} &&\dfrac{df}{dx} &= n(\text{that stuff})^{n-1} \cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\left(u^n \right) = n u^{n-1} \cdot \dfrac{du}{dx}$$

The following problems illustrate and let you practice.

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Chain Rule and Exponentials

\begin{align*}

\text{If} && f(x) &= e^{\text{(stuff)}}, \\[8px]
\text{then} &&\dfrac{df}{dx} &= e^{\text{(that stuff)}}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}e^u = e^u \cdot \dfrac{du}{dx}$$

The following problems illustrate and let you practice.

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Chain Rule and Trig Functions

\begin{align*}

\text{If} && f(x) &= \sin\text{(stuff)}, \\[8px]
\text{then} &&\dfrac{df}{dx} &= \cos\text{(that stuff)}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\sin u = \cos u \cdot \dfrac{du}{dx}$$

\begin{align*}

\text{If} && f(x) &= \cos\text{(stuff)}, \\[8px] \text{then} &&\dfrac{df}{dx} &= -\sin\text{(that stuff)}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\cos u = -\sin u \cdot \dfrac{du}{dx}$$

\begin{align*}

\text{If} && f(x) &= \tan\text{(stuff)}, \\[8px] \text{then} &&\dfrac{df}{dx} &= \sec^2\text{(that stuff)}\cdot \dfrac{d}{dx}(\text{that stuff})

\end{align*}

You’ll usually see this written as

$$\dfrac{d}{dx}\tan u = \sec^2 u \cdot \dfrac{du}{dx}$$

The following problems illustrate and let you practice.

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Chain Rule and Product or Quotient Rule

The next few problems require using the Chain rule with the Product rule or with the Quotient rule.

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Other Routine Chain Rule Problems

We’ll end this screen with some other typical Chain Rule problems you’re likely to encounter on an exam — including a few from actual university exams, which we hope will seem routine after the work you’ve done above. If they don’t seem routine yet, they will soon, as long as you keep practicing!

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This next problem is a little different, since you’re given $f'(x)$ and then asked to find the derivative of $f(x^2).$ You may encounter a similar problem in your homework or on an exam; this problem was taken, in fact, from an exam at a well-known university.

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On the next screen we’ll introduce problems that require using the Chain Rule more than once. It’s a small step from what we’ve done above. Since for the rest of the course you’re going to need to take such derivatives quickly and correctly, please proceed there to practice as soon as you can.

Do you have questions about any of the problems on this screen, or other Chain Rule problems you’re working on? If you post on the Forum, we’ll do our best to assist!