Calculus Examples

$y = (\cos (5 x (\sin (5 x))))$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 5 x (\sin (5 x))$ .

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Step 1.1

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

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

Step 1.2

The derivative of $\cos (u_{1})$ with respect to $u_{1}$ is $- \sin (u_{1})$ .

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

Step 1.3

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

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

Step 2

Differentiate using the Constant Multiple Rule.

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Step 2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x \sin (5 x)$ with respect to $x$ is $5 \frac{d}{d x} [x \sin (5 x)]$ .

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

Step 2.2

Multiply $5$ by $- 1$ .

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

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \sin (5 x)$ .

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

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 $g (x) = 5 x$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 5

Differentiate.

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Step 5.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

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

Step 5.2

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

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

Step 5.3

Simplify the expression.

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Step 5.3.1

Multiply $5$ by $1$ .

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

Step 5.3.2

Move $5$ to the left of $\cos (5 x)$ .

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

Step 5.4

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

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

Step 5.5

Multiply $\sin (5 x)$ by $1$ .

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$- 5 \sin (5 x \sin (5 x)) (x (5 \cos (5 x))) - 5 \sin (5 x \sin (5 x)) \sin (5 x)$

Step 6.2

Multiply $5$ by $- 5$ .

$- 25 \sin (5 x \sin (5 x)) x \cos (5 x) - 5 \sin (5 x \sin (5 x)) \sin (5 x)$

Step 6.3

Reorder terms.

$- 25 x \cos (5 x) \sin (5 x \sin (5 x)) - 5 \sin (5 x) \sin (5 x \sin (5 x))$