Calculus Examples

$y = x^{3} \cos (x^{2})$

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 $f (x) = x^{3}$ and $g (x) = \cos (x^{2})$ .

$x^{3} \frac{d}{d x} [\cos (x^{2})] + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

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

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Step 2.1

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

$x^{3} (\frac{d}{d u} [\cos (u)] \frac{d}{d x} [x^{2}]) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 2.2

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

$x^{3} (- \sin (u) \frac{d}{d x} [x^{2}]) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 2.3

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

$x^{3} (- \sin (x^{2}) \frac{d}{d x} [x^{2}]) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 3

Differentiate using the Power Rule.

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Step 3.1

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

$x^{3} (- \sin (x^{2}) (2 x)) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 3.2

Multiply $2$ by $- 1$ .

$x^{3} (- 2 \sin (x^{2}) x) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 4

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 4.1

Move $x$ .

$x \cdot x^{3} (- 2 \sin (x^{2})) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 4.2

Multiply $x$ by $x^{3}$ .

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Step 4.2.1

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

$x^{1} x^{3} (- 2 \sin (x^{2})) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 4.2.2

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

$x^{1 + 3} (- 2 \sin (x^{2})) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 4.3

Add $1$ and $3$ .

$x^{4} (- 2 \sin (x^{2})) + \cos (x^{2}) \frac{d}{d x} [x^{3}]$

Step 5

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{4} (- 2 \sin (x^{2})) + \cos (x^{2}) (3 x^{2})$

Step 6

Reorder terms.

$- 2 x^{4} \sin (x^{2}) + 3 x^{2} \cos (x^{2})$