Calculus Examples

$\tan^{2} (n θ)$

Step 1

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

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as $\tan (n θ)$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d θ} [\tan (n θ)]$

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 $n = 2$ .

$2 u_{1} \frac{d}{d θ} [\tan (n θ)]$

Step 1.3

Replace all occurrences of $u_{1}$ with $\tan (n θ)$ .

$2 \tan (n θ) \frac{d}{d θ} [\tan (n θ)]$

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $n θ$ .

$2 \tan (n θ) (\frac{d}{d u_{2}} [\tan (u_{2})] \frac{d}{d θ} [n θ])$

Step 2.2

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

$2 \tan (n θ) (\sec^{2} (u_{2}) \frac{d}{d θ} [n θ])$

Step 2.3

Replace all occurrences of $u_{2}$ with $n θ$ .

$2 \tan (n θ) (\sec^{2} (n θ) \frac{d}{d θ} [n θ])$

Step 3

Differentiate.

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Step 3.1

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

$2 \tan (n θ) \sec^{2} (n θ) (n \frac{d}{d θ} [θ])$

Step 3.2

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

$2 \tan (n θ) \sec^{2} (n θ) (n \cdot 1)$

Step 3.3

Simplify the expression.

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Step 3.3.1

Multiply $n$ by $1$ .

$2 \tan (n θ) \sec^{2} (n θ) n$

Step 3.3.2

Reorder the factors of $2 \tan (n θ) \sec^{2} (n θ) n$ .

$2 n \sec^{2} (n θ) \tan (n θ)$