Calculus Examples

$y = \sin (\tan (3 x))$

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

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

$\frac{d}{d u_{1}} [\sin (u_{1})] \frac{d}{d x} [\tan (3 x)]$

The derivative of $\sin (u_{1})$ with respect to $u_{1}$ is $\cos (u_{1})$ .

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

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

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

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 $g (x) = 3 x$ .

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

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

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

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

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

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

Differentiate.

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Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

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

$\cos (\tan (3 x)) (\sec^{2} (3 x) (3 \cdot 1))$

Simplify the expression.

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Multiply $3$ by $1$ .

$\cos (\tan (3 x)) (\sec^{2} (3 x) \cdot 3)$

Move $3$ to the left of $\sec^{2} (3 x)$ .

$\cos (\tan (3 x)) (3 \cdot \sec^{2} (3 x))$

Simplify.

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Move $3$ to the left of $\cos (\tan (3 x))$ .

$3 \cos (\tan (3 x)) \sec^{2} (3 x)$

Reorder the factors of $3 \cos (\tan (3 x)) \sec^{2} (3 x)$ .

$3 \sec^{2} (3 x) \cos (\tan (3 x))$

Rewrite $\sec (3 x)$ in terms of sines and cosines.

$3 {(\frac{1}{\cos (3 x)})}^{2} \cos (\tan (3 x))$

Apply the product rule to $\frac{1}{\cos (3 x)}$ .

$3 \frac{1^{2}}{\cos^{2} (3 x)} \cos (\tan (3 x))$

One to any power is one.

$3 \frac{1}{\cos^{2} (3 x)} \cos (\tan (3 x))$

Combine $3$ and $\frac{1}{\cos^{2} (3 x)}$ .

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

Combine $\frac{3}{\cos^{2} (3 x)}$ and $\cos (\tan (3 x))$ .

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

Factor $\cos (3 x)$ out of $\cos^{2} (3 x)$ .

$\frac{3 \cos (\tan (3 x))}{\cos (3 x) \cos (3 x)}$

Separate fractions.

$\frac{3}{\cos (3 x)} \cdot \frac{\cos (\tan (3 x))}{\cos (3 x)}$

Rewrite $\frac{\cos (\tan (3 x))}{\cos (3 x)}$ as a product.

$\frac{3}{\cos (3 x)} (\cos (\tan (3 x)) \frac{1}{\cos (3 x)})$

Write $\cos (\tan (3 x))$ as a fraction with denominator $1$ .

$\frac{3}{\cos (3 x)} (\frac{\cos (\tan (3 x))}{1} \cdot \frac{1}{\cos (3 x)})$

Simplify.

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Divide $\cos (\tan (3 x))$ by $1$ .

$\frac{3}{\cos (3 x)} (\cos (\tan (3 x)) \frac{1}{\cos (3 x)})$

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

$\frac{3}{\cos (3 x)} (\cos (\tan (3 x)) \sec (3 x))$

Separate fractions.

$\frac{3}{1} \cdot \frac{1}{\cos (3 x)} (\cos (\tan (3 x)) \sec (3 x))$

Convert from $\frac{1}{\cos (3 x)}$ to $\sec (3 x)$ .

$\frac{3}{1} \sec (3 x) (\cos (\tan (3 x)) \sec (3 x))$

Divide $3$ by $1$ .

$3 \sec (3 x) (\cos (\tan (3 x)) \sec (3 x))$

Multiply $3 \sec (3 x) (\cos (\tan (3 x)) \sec (3 x))$ .

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Raise $\sec (3 x)$ to the power of $1$ .

$3 (\sec^{1} (3 x) \sec (3 x)) \cos (\tan (3 x))$

Raise $\sec (3 x)$ to the power of $1$ .

$3 (\sec^{1} (3 x) \sec^{1} (3 x)) \cos (\tan (3 x))$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$3 {\sec (3 x)}^{1 + 1} \cos (\tan (3 x))$

Add $1$ and $1$ .

$3 \sec^{2} (3 x) \cos (\tan (3 x))$