Calculus Examples

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

Step 1

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

Tap for more steps...

Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

Differentiate.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Simplify the expression.

Tap for more steps...

Step 2.3.1

Move $2$ to the left of $a$ .

$- \sin (a x^{2}) (2 \cdot a x)$

Step 2.3.2

Multiply $2$ by $- 1$ .

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

Step 2.3.3

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

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