FUNCTION OF A FUNCTION RULE OF DIFFERENTIATION (CHAIN RULE)

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It is often easier to make a substitution before differentiating. For instance, if y is a function of x, then

$$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$

This is known as the function of function rule or chain rule.

Example 1

Differentiate $y=(x^3-1)^{100}$

Solution:

We say let $u=(x^3-1)$ so that $y=u^{100}$

Differentiating using the power rule

$\frac{dy}{du}=100u^{99}$, $\frac{du}{dx}=3x^2$

Recalled that

$$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$

$$\frac{dy}{dx}=100u^{99}\times 3x^2$$

Recalled that $u=x^3-1$

$$\frac{dy}{dx}=100(x^3-1)^{99}\times 3x^2$$

$$\frac{dy}{dx}=300x^2(x^2-1)^{99}$$

Example 2

Differentiate $y=(5x+4)^2$ using chain rule

Solution:

Let $u=(5+4)^2$ so that $y=u^2$

$\frac{dy}{du}=2u$ and $\frac{du}{dx}=5$

Recalled that

$$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$

$$\frac{dy}{dx}=2u ×5$$

$$\frac{dy}{dx}=(2(5x+4))\times 5$$

$$\frac{dy}{dx}=(10x+8)(5)$$

$$\frac{dy}{dx}=50x+40$$

Example 3

Differentiate $y=\cos^3 x$

Solution:

$\cos^3x$ is the same as $(\cos x)^3$

Let $u=\cos x$, so that $y=u^3$

$\frac{dy}{du}=3u^2$, $\frac{du}{dx}=-\sin x$

$$\frac{dy}{dx}=3u^2\times-\sin x$$

$$\frac{dy}{dx}=3(\cos x)^2\times-\sin x$$

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

$$\frac{dy}{dx}=-3\sin x\cos^2x$$

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Example 4

If $y=(8x+\frac{1}{2})^2$, Evaluate $\frac{dy}{dx}$

Solution:

Let $u=8x+\frac{1}{2}$, so that $y=u^3$

$\frac{dy}{du}=2u$, $\frac{du}{dx}=8$

$$\frac{dy}{dx}=2u\times 8$$

Recalled that $u=8x+\frac{1}{2}$

$$\frac{dy}{dx}=2(8x+\frac{1}{2})\times 8$$

$$\frac{dy}{dx}=(16x+\frac{2}{2})8$$

$$\frac{dy}{dx}=(16x+1)8$$

$$\frac{dy}{dx}=128x+8$$

There you have it. We will be looking at the differentiation of trigonometric function in this post

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