Calculus Examples

$y = \tan (2 x)$

Step 1

Differentiate both sides of the equation.

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

Step 2

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

$y'$

Step 3

Differentiate the right side of the equation.

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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 $g (x) = 2 x$ .

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

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

$\sec^{2} (2 x) (2 \cdot 1)$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $2$ by $1$ .

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

Step 3.2.3.2

Move $2$ to the left of $\sec^{2} (2 x)$ .

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

Step 4

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

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

Step 5

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

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