Calculus Examples

Find the derivative of $\sin^{2} (x + 5)$

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 $g (x) = \sin (x + 5)$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\sin (x + 5)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [\sin (x + 5)]$

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 $n = 2$ .

$2 u_{1} \frac{d}{d x} [\sin (x + 5)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\sin (x + 5)$ .

$2 \sin (x + 5) \frac{d}{d x} [\sin (x + 5)]$

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 $g (x) = x + 5$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x + 5$ .

$2 \sin (x + 5) (\frac{d}{d u_{2}} [\sin (u_{2})] \frac{d}{d x} [x + 5])$

Step 2.2

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

$2 \sin (x + 5) (\cos (u_{2}) \frac{d}{d x} [x + 5])$

Step 2.3

Replace all occurrences of $u_{2}$ with $x + 5$ .

$2 \sin (x + 5) (\cos (x + 5) \frac{d}{d x} [x + 5])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$2 \sin (x + 5) \cos (x + 5) (\frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 3.2

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

$2 \sin (x + 5) \cos (x + 5) (1 + \frac{d}{d x} [5])$

Step 3.3

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$2 \sin (x + 5) \cos (x + 5) (1 + 0)$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $1$ and $0$ .

$2 \sin (x + 5) \cos (x + 5) \cdot 1$

Step 3.4.2

Multiply $2$ by $1$ .

$2 \sin (x + 5) \cos (x + 5)$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $2 \sin (x + 5) \cos (x + 5)$ .

$2 \cos (x + 5) \sin (x + 5)$

Step 4.2

Reorder $2 \cos (x + 5)$ and $\sin (x + 5)$ .

$\sin (x + 5) (2 \cos (x + 5))$

Step 4.3

Reorder $\sin (x + 5)$ and $2$ .

$2 \cdot \sin (x + 5) \cos (x + 5)$

Step 4.4

Apply the sine double-angle identity.

$\sin (2 (x + 5))$

Step 4.5

Apply the distributive property.

$\sin (2 x + 2 \cdot 5)$

Step 4.6

Multiply $2$ by $5$ .

$\sin (2 x + 10)$