Calculus Examples

$\sin^{2} (3 - x)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (3 - x)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (3 - x)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (3 - x)]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 2$ .

$2 u_{1} \frac{d}{d x} [\sin (3 - x)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (3 - x)$ .

$2 \sin (3 - x) \frac{d}{d x} [\sin (3 - x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = 3 - x$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $3 - x$ .

$2 \sin (3 - x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [3 - x])$

Step 2.2

The derivative of $\sin (u_{2})$ with respect to $u_{2}$ is $\cos (u_{2})$ .

$2 \sin (3 - x) (\cos (u_{2}) \frac{d}{d x} [3 - x])$

Step 2.3

Replace all occurrences of $u_{2}$ with $3 - x$ .

$2 \sin (3 - x) (\cos (3 - x) \frac{d}{d x} [3 - x])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 - x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- x]$ .

$2 \sin (3 - x) \cos (3 - x) (\frac{d}{d x} [3] + \frac{d}{d x} [- x])$

Step 3.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 \sin (3 - x) \cos (3 - x) (0 + \frac{d}{d x} [- x])$

Step 3.3

Add $0$ and $\frac{d}{d x} [- x]$ .

$2 \sin (3 - x) \cos (3 - x) \frac{d}{d x} [- x]$

Step 3.4

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$2 \sin (3 - x) \cos (3 - x) (- \frac{d}{d x} [x])$

Step 3.5

Multiply $- 1$ by $2$ .

$- 2 \sin (3 - x) \cos (3 - x) \frac{d}{d x} [x]$

Step 3.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- 2 \sin (3 - x) \cos (3 - x) \cdot 1$

Step 3.7

Simplify the expression.

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Step 3.7.1

Multiply $- 2$ by $1$ .

$- 2 \sin (3 - x) \cos (3 - x)$

Step 3.7.2

Reorder the factors of $- 2 \sin (3 - x) \cos (3 - x)$ .

$- 2 \cos (3 - x) \sin (3 - x)$