Calculus Examples

$y = \tan (4 x^{2})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\tan (4 x^{2}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \tan (x)$ and $g (x) = 4 x^{2}$ .

Tap for more steps...

Step 3.1.1

To apply the Chain Rule, set $u$ as $4 x^{2}$ .

$\frac{d}{d u} [\tan (u)] \frac{d}{d x} [4 x^{2}]$

Step 3.1.2

The derivative of $\tan (u)$ with respect to $u$ is $\sec^{2} (u)$ .

$\sec^{2} (u) \frac{d}{d x} [4 x^{2}]$

Step 3.1.3

Replace all occurrences of $u$ with $4 x^{2}$ .

$\sec^{2} (4 x^{2}) \frac{d}{d x} [4 x^{2}]$

Step 3.2

Differentiate.

Tap for more steps...

Step 3.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$\sec^{2} (4 x^{2}) (4 \frac{d}{d x} [x^{2}])$

Step 3.2.2

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

$\sec^{2} (4 x^{2}) (4 (2 x))$

Step 3.2.3

Simplify the expression.

Tap for more steps...

Step 3.2.3.1

Multiply $2$ by $4$ .

$\sec^{2} (4 x^{2}) (8 x)$

Step 3.2.3.2

Reorder the factors of $\sec^{2} (4 x^{2}) (8 x)$ .

$8 x \sec^{2} (4 x^{2})$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 8 x \sec^{2} (4 x^{2})$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 8 x \sec^{2} (4 x^{2})$