Calculus Examples

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

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

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

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

Step 3

Raise $\cos (x^{2})$ to the power of $1$ .

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

Step 4

Raise $\cos (x^{2})$ to the power of $1$ .

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

Step 5

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

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

Step 6

Differentiate using the Power Rule.

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Step 6.1

Add $1$ and $1$ .

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

Step 6.2

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

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

Step 7

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 $g (x) = x^{2}$ .

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Step 7.1

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

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

Step 7.2

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

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

Step 7.3

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

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

Step 8

Raise $\sin (x^{2})$ to the power of $1$ .

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

Step 9

Raise $\sin (x^{2})$ to the power of $1$ .

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

Step 10

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

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

Step 11

Add $1$ and $1$ .

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

Step 12

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

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

Step 13

Simplify the expression.

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Step 13.1

Multiply $2$ by $- 1$ .

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

Step 13.2

Reorder terms.

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