Calculus Examples

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

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \cos (x^{3})$ .

Tap for more steps...

Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\cos (x^{3})$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u_{1}$ with $\cos (x^{3})$ .

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

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 $g (x) = x^{3}$ .

Tap for more steps...

Step 2.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

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

Step 3

Differentiate using the Power Rule.

Tap for more steps...

Step 3.1

Multiply $- 1$ by $2$ .

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

Step 3.2

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

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

Step 3.3

Simplify the expression.

Tap for more steps...

Step 3.3.1

Multiply $3$ by $- 2$ .

$- 6 \cos (x^{3}) \sin (x^{3}) x^{2}$

Step 3.3.2

Reorder the factors of $- 6 \cos (x^{3}) \sin (x^{3}) x^{2}$ .

$- 6 x^{2} \cos (x^{3}) \sin (x^{3})$