Calculus Examples

$y = 15 x \cos (x)$

Step 1

Find the first derivative.

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Step 1.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x \cos (x)$ with respect to $x$ is $15 \frac{d}{d x} [x \cos (x)]$ .

$15 \frac{d}{d x} [x \cos (x)]$

Step 1.2

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 $g (x) = \cos (x)$ .

$15 (x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x])$

Step 1.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$15 (x (- \sin (x)) + \cos (x) \frac{d}{d x} [x])$

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

$15 (x (- \sin (x)) + \cos (x) \cdot 1)$

Step 1.4.2

Multiply $\cos (x)$ by $1$ .

$15 (x (- \sin (x)) + \cos (x))$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$15 (x (- \sin (x))) + 15 \cos (x)$

Step 1.5.2

Multiply $- 1$ by $15$ .

$f' (x) = - 15 x \sin (x) + 15 \cos (x)$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 15 x \sin (x) + 15 \cos (x)$ with respect to $x$ is $\frac{d}{d x} [- 15 x \sin (x)] + \frac{d}{d x} [15 \cos (x)]$ .

$\frac{d}{d x} [- 15 x \sin (x)] + \frac{d}{d x} [15 \cos (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [- 15 x \sin (x)]$ .

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Step 2.2.1

Since $- 15$ is constant with respect to $x$ , the derivative of $- 15 x \sin (x)$ with respect to $x$ is $- 15 \frac{d}{d x} [x \sin (x)]$ .

$- 15 \frac{d}{d x} [x \sin (x)] + \frac{d}{d x} [15 \cos (x)]$

Step 2.2.2

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

$- 15 (x \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x]) + \frac{d}{d x} [15 \cos (x)]$

Step 2.2.3

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$- 15 (x \cos (x) + \sin (x) \frac{d}{d x} [x]) + \frac{d}{d x} [15 \cos (x)]$

Step 2.2.4

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

$- 15 (x \cos (x) + \sin (x) \cdot 1) + \frac{d}{d x} [15 \cos (x)]$

Step 2.2.5

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

$- 15 (x \cos (x) + \sin (x)) + \frac{d}{d x} [15 \cos (x)]$

Step 2.3

Evaluate $\frac{d}{d x} [15 \cos (x)]$ .

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Step 2.3.1

Since $15$ is constant with respect to $x$ , the derivative of $15 \cos (x)$ with respect to $x$ is $15 \frac{d}{d x} [\cos (x)]$ .

$- 15 (x \cos (x) + \sin (x)) + 15 \frac{d}{d x} [\cos (x)]$

Step 2.3.2

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$- 15 (x \cos (x) + \sin (x)) + 15 (- \sin (x))$

Step 2.3.3

Multiply $- 1$ by $15$ .

$- 15 (x \cos (x) + \sin (x)) - 15 \sin (x)$

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

$- 15 (x \cos (x)) - 15 \sin (x) - 15 \sin (x)$

Step 2.4.2

Subtract $15 \sin (x)$ from $- 15 \sin (x)$ .

$f'' (x) = - 15 x \cos (x) - 30 \sin (x)$