Differentiation Rules

**fantasylo** · November 18th 2012, 11:21 PM

Hi, I do not understand the rule 5. Is it possible to include an example for me to understand better?

Thanks.

**Prove It** · November 18th 2012, 11:30 PM

That's the chain rule. It's easier to write it like this: $\displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \end{align*}$ . Basically it means that if you have a composition of functions, you need to isolate the "inner" function (u) and work out its derivative, then write y as a function of u and evaluate its derivative. After converting it back to a function of x, multiply the two derivatives together to get the total derivative.

**MarkFL** · November 18th 2012, 11:44 PM

Here are a couple of examples:

a) $y=\sin(x^3)$

$\frac{dy}{dx}=\cos(x^3)\frac{d}{dx}(x^3)=3x^2\cos( x^3)$

b) $y=e^{\tan(2x)}$

$\frac{dy}{dx}=e^{\tan(2x)}\frac{d}{dx}(\tan(2x))=e ^{\tan(2x)}\sec^2(2x)\frac{d}{dx}(2x)=2\sec^2(2x)e ^{\tan(2x)}$

**Prove It** · November 19th 2012, 08:00 AM

Originally Posted by MarkFL2

Here are a couple of examples:

a) $y=\sin(x^3)$

$\frac{dy}{dx}=\cos(x^3)\frac{d}{dx}(x^3)=3x^2\cos( x^3)$

b) $y=e^{\tan(2x)}$

$\frac{dy}{dx}=e^{\tan(2x)}\frac{d}{dx}(\tan(2x))=e ^{\tan(2x)}\sec^2(2x)\frac{d}{dx}(2x)=2\sec^2(2x)e ^{\tan(2x)}$

Or to use the easier-to-remember form of the chain rule that I posted...

1. $\displaystyle \begin{align*} y = \sin{\left( x^3 \right)} \end{align*}$

Let $\displaystyle \begin{align*} u = x^3 \end{align*}$ which gives $\displaystyle \begin{align*} y = \sin{u} \end{align*}$ .

Then $\displaystyle \begin{align*} \frac{du}{dx} = 3x^2 \end{align*}$ and $\displaystyle \begin{align*} \frac{dy}{du} = \cos{u} = \cos{\left(x^3 \right)} \end{align*}$ .

So $\displaystyle \begin{align*} \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 3x^2\cos{\left(x^3 \right)} \end{align*}$

See if you can use a similar process with the second example. You will need to use the Chain Rule TWICE.

**fantasylo** · November 19th 2012, 09:46 PM

Okay. I got it. Thank you Prove It and MarkFL2 for your clear examples and explanation!!

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