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# 3.13: Multirule Derivatives

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Okay, let’s talk about $$\frac{d}{dx} \ e^{x^2 + x} \sin(x)$$. If you’re thinking this looks like a product rule, but it also looks like a chain rule, you’re right. To compute this derivative, we need to do the chain rule and the product rule. This is because it is a multirule problem. Let’s do this example

## Multirule

Compute $$\frac{d}{dx} \ e^{x^2 + x} \sin(x)$$.

The way I like to break this down is to consider a little rule and a big rule. In this case, the little rule is the chain rule problem $$\frac{d}{dx} \ e^{x^2 + x}$$. If we do this problem, we see that $$f = e^x$$, $$f' = e^x$$, $$g = x^2 + x$$ and $$g' = 2x + 1$$. So we have

$\begin{equation*} \frac{d}{dx} \ e^{x^2 + x} = {\color{red} e^{x^2 + x} (2x + 1)}. \end{equation*}$

Now we are ready to do the big rule, which is the product rule. At this point we go back to the original problem $$\frac{d}{dx} \ e^{x^2 + x} \sin(x)$$. For this product rule, we see $$f = e^{x^2 + x}$$, $$g = \sin(x)$$, $$g' = \cos(x)$$. What is $${\color{red} f'}$$? Why, that’s what we just computed in the equation above! So $${\color{red} f' = e^{x^2 + x}(2x+1)}$$. Putting this all together with the product rule $$f g' + g f'$$, we have

\begin{align*} \frac{d}{dx} \ e^{x^2 + x} \sin(x) & = f g' + g {\color{red} f'} \\ & = \boxed{e^{x^2 + x} \cos(x) + \sin(x) {\color{red} e^{x^2 + x} (2x + 1)}} . \end{align*}

## Multirule

Compute $$\frac{d}{dx} \frac{x}{\sin(x^2 + x)}$$.

little chain rule: $$\frac{d}{dx} \sin(x^2 + x)$$

$$\begin{array}{ll} f = \sin(x) & g = x^2 + x \\ f' = \cos(x) & g' = 2x + 1 \end{array}$$

Result: $${\color{red} \cos(x^2 + x) \cdot (2x+1)}$$

Big quotient rule (aka the whole problem):$$\frac{d}{dx} \frac{x}{\sin(x^2 + x)}$$

$$\begin{array}{ll} f = x & g = \sin(x^2 + x) \\ f' = 1 & g' = {\color{red} \cos(x^2 + x) \cdot (2x+1)} \end{array}$$

Result: $$\boxed{\frac{\sin(x^2 + x) \cdot 1 - x \cos(x^2 + x) \cdot (2x+1)}{(\sin(x^2+x))^2}}$$

This page titled 3.13: Multirule Derivatives is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Tyler Seacrest via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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