Calculus Examples

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

Step 1

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

$4 \frac{d}{d x} [x^{2} \cos (4 x)]$

Step 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 $g (x) = \cos (4 x)$ .

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

Step 3

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 $g (x) = 4 x$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $4 x$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $4 x$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

Multiply $4$ by $- 1$ .

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

Step 4.3

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

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

Step 4.4

Multiply $- 4$ by $1$ .

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

Step 4.5

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

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Combine terms.

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Step 5.2.1

Multiply $- 4$ by $4$ .

$- 16 (x^{2} (\sin (4 x))) + 4 (\cos (4 x) (2 x))$

Step 5.2.2

Multiply $2$ by $4$ .

$- 16 x^{2} \sin (4 x) + 8 \cos (4 x) x$

Step 5.3

Reorder terms.

$- 16 x^{2} \sin (4 x) + 8 x \cos (4 x)$