Calculus Examples

$2 \sin^{2} (1 + x)$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 \sin^{2} (1 + x)$ with respect to $x$ is $2 \frac{d}{d x} [\sin^{2} (1 + x)]$ .

$2 \frac{d}{d x} [\sin^{2} (1 + x)]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sin (1 + x)$ .

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

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

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

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

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

$2 (2 \sin (1 + x) \frac{d}{d x} [\sin (1 + x)])$

Step 3

Multiply $2$ by $2$ .

$4 (\sin (1 + x) \frac{d}{d x} [\sin (1 + x)])$

Step 4

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 $g (x) = 1 + x$ .

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To apply the Chain Rule, set $u_{2}$ as $1 + x$ .

$4 \sin (1 + x) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [1 + x])$

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

$4 \sin (1 + x) (\cos (u_{2}) \frac{d}{d x} [1 + x])$

Replace all occurrences of $u_{2}$ with $1 + x$ .

$4 \sin (1 + x) (\cos (1 + x) \frac{d}{d x} [1 + x])$

Step 5

Differentiate.

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By the Sum Rule, the derivative of $1 + x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x]$ .

$4 \sin (1 + x) \cos (1 + x) (\frac{d}{d x} [1] + \frac{d}{d x} [x])$

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$4 \sin (1 + x) \cos (1 + x) (0 + \frac{d}{d x} [x])$

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

$4 \sin (1 + x) \cos (1 + x) \frac{d}{d x} [x]$

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

$4 \sin (1 + x) \cos (1 + x) \cdot 1$

Simplify the expression.

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Multiply $4$ by $1$ .

$4 \sin (1 + x) \cos (1 + x)$

Reorder the factors of $4 \sin (1 + x) \cos (1 + x)$ .

$4 \cos (1 + x) \sin (1 + x)$