Calculus Examples

$y = x \cos (5 x^{2})$

Step 1

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 $g (x) = \cos (5 x^{2})$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \cos (x)$ and $g (x) = 5 x^{2}$ .

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Differentiate.

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{2}$ with respect to $x$ is $5 \frac{d}{d x} [x^{2}]$ .

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

Step 3.2

Multiply $5$ by $- 1$ .

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $- 5$ .

$x (- 10 \sin (5 x^{2}) x) + \cos (5 x^{2}) \frac{d}{d x} [x]$

Step 4

Raise $x$ to the power of $1$ .

$x^{1} x (- 10 \sin (5 x^{2})) + \cos (5 x^{2}) \frac{d}{d x} [x]$

Step 5

Raise $x$ to the power of $1$ .

$x^{1} x^{1} (- 10 \sin (5 x^{2})) + \cos (5 x^{2}) \frac{d}{d x} [x]$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{1 + 1} (- 10 \sin (5 x^{2})) + \cos (5 x^{2}) \frac{d}{d x} [x]$

Step 7

Add $1$ and $1$ .

$x^{2} (- 10 \sin (5 x^{2})) + \cos (5 x^{2}) \frac{d}{d x} [x]$

Step 8

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

$x^{2} (- 10 \sin (5 x^{2})) + \cos (5 x^{2}) \cdot 1$

Step 9

Simplify the expression.

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Step 9.1

Multiply $\cos (5 x^{2})$ by $1$ .

$x^{2} (- 10 \sin (5 x^{2})) + \cos (5 x^{2})$

Step 9.2

Reorder terms.

$- 10 x^{2} \sin (5 x^{2}) + \cos (5 x^{2})$