Calculus Examples

$y = 4 x^{2} \sin (x) + 8 x \cos (x) - 8 \sin (x)$

Step 1

By the Sum Rule, the derivative of $4 x^{2} \sin (x) + 8 x \cos (x) - 8 \sin (x)$ with respect to $x$ is $\frac{d}{d x} [4 x^{2} \sin (x)] + \frac{d}{d x} [8 x \cos (x)] + \frac{d}{d x} [- 8 \sin (x)]$ .

$\frac{d}{d x} [4 x^{2} \sin (x)] + \frac{d}{d x} [8 x \cos (x)] + \frac{d}{d x} [- 8 \sin (x)]$

Step 2

Evaluate $\frac{d}{d x} [4 x^{2} \sin (x)]$ .

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Step 2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2} \sin (x)$ with respect to $x$ is $4 \frac{d}{d x} [x^{2} \sin (x)]$ .

$4 \frac{d}{d x} [x^{2} \sin (x)] + \frac{d}{d x} [8 x \cos (x)] + \frac{d}{d x} [- 8 \sin (x)]$

Step 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 $f (x) = x^{2}$ and $g (x) = \sin (x)$ .

$4 (x^{2} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [8 x \cos (x)] + \frac{d}{d x} [- 8 \sin (x)]$

Step 2.3

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

$4 (x^{2} \cos (x) + \sin (x) \frac{d}{d x} [x^{2}]) + \frac{d}{d x} [8 x \cos (x)] + \frac{d}{d x} [- 8 \sin (x)]$

Step 2.4

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

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + \frac{d}{d x} [8 x \cos (x)] + \frac{d}{d x} [- 8 \sin (x)]$

Step 3

Evaluate $\frac{d}{d x} [8 x \cos (x)]$ .

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x \cos (x)$ with respect to $x$ is $8 \frac{d}{d x} [x \cos (x)]$ .

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + 8 \frac{d}{d x} [x \cos (x)] + \frac{d}{d x} [- 8 \sin (x)]$

Step 3.2

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

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + 8 (x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 8 \sin (x)]$

Step 3.3

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

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + 8 (x (- \sin (x)) + \cos (x) \frac{d}{d x} [x]) + \frac{d}{d x} [- 8 \sin (x)]$

Step 3.4

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

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + 8 (x (- \sin (x)) + \cos (x) \cdot 1) + \frac{d}{d x} [- 8 \sin (x)]$

Step 3.5

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

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + 8 (x (- \sin (x)) + \cos (x)) + \frac{d}{d x} [- 8 \sin (x)]$

Step 4

Evaluate $\frac{d}{d x} [- 8 \sin (x)]$ .

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Step 4.1

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8 \sin (x)$ with respect to $x$ is $- 8 \frac{d}{d x} [\sin (x)]$ .

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + 8 (x (- \sin (x)) + \cos (x)) - 8 \frac{d}{d x} [\sin (x)]$

Step 4.2

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

$4 (x^{2} \cos (x) + \sin (x) (2 x)) + 8 (x (- \sin (x)) + \cos (x)) - 8 \cos (x)$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 (x^{2} \cos (x)) + 4 (\sin (x) (2 x)) + 8 (x (- \sin (x)) + \cos (x)) - 8 \cos (x)$

Step 5.2

Apply the distributive property.

$4 (x^{2} \cos (x)) + 4 (\sin (x) (2 x)) + 8 (x (- \sin (x))) + 8 \cos (x) - 8 \cos (x)$

Step 5.3

Combine terms.

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Step 5.3.1

Multiply $2$ by $4$ .

$4 x^{2} \cos (x) + 8 (\sin (x) (x)) + 8 (x (- \sin (x))) + 8 \cos (x) - 8 \cos (x)$

Step 5.3.2

Multiply $- 1$ by $8$ .

$4 x^{2} \cos (x) + 8 \sin (x) x - 8 (x (\sin (x))) + 8 \cos (x) - 8 \cos (x)$

Step 5.3.3

Subtract $8 x \sin (x)$ from $8 \sin (x) x$ .

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Step 5.3.3.1

Move $\sin (x)$ .

$4 x^{2} \cos (x) + 8 x \sin (x) - 8 x \sin (x) + 8 \cos (x) - 8 \cos (x)$

Step 5.3.3.2

Subtract $8 x \sin (x)$ from $8 x \sin (x)$ .

$4 x^{2} \cos (x) + 0 + 8 \cos (x) - 8 \cos (x)$

Step 5.3.4

Add $4 x^{2} \cos (x)$ and $0$ .

$4 x^{2} \cos (x) + 8 \cos (x) - 8 \cos (x)$

Step 5.3.5

Subtract $8 \cos (x)$ from $8 \cos (x)$ .

$4 x^{2} \cos (x) + 0$

Step 5.3.6

Add $4 x^{2} \cos (x)$ and $0$ .

$4 x^{2} \cos (x)$