Find the Derivative – d/dx (sin(2x))/(cos(x))

$\frac{\sin (2 x)}{\cos (x)}$

Rewrite $\frac{\sin (2 x)}{\cos (x)}$ as a product.

$\frac{d}{d x} [\sin (2 x) \frac{1}{\cos (x)}]$

Write $\sin (2 x)$ as a fraction with denominator $1$ .

$\frac{d}{d x} [\frac{\sin (2 x)}{1} \cdot \frac{1}{\cos (x)}]$

Simplify.

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Divide $\sin (2 x)$ by $1$ .

$\frac{d}{d x} [\sin (2 x) \frac{1}{\cos (x)}]$

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

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

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sin (2 x)$ and $g (x) = \sec (x)$ .

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

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\sin (2 x) (\sec (x) \tan (x)) + \sec (x) \frac{d}{d x} [\sin (2 x)]$

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 $g (x) = 2 x$ .

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

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

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

$\sin (2 x) \sec (x) \tan (x) + \sec (x) (\cos (u) \frac{d}{d x} [2 x])$

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

$\sin (2 x) \sec (x) \tan (x) + \sec (x) (\cos (2 x) \frac{d}{d x} [2 x])$

Differentiate.

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

$\sin (2 x) \sec (x) \tan (x) + \sec (x) \cos (2 x) (2 \frac{d}{d x} [x])$

Move $2$ to the left of $\sec (x) \cos (2 x)$ .

$\sin (2 x) \sec (x) \tan (x) + 2 \cdot (\sec (x) \cos (2 x)) \frac{d}{d x} [x]$

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

$\sin (2 x) \sec (x) \tan (x) + 2 \sec (x) \cos (2 x) \cdot 1$

Simplify the expression.

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Multiply $2$ by $1$ .

$\sin (2 x) \sec (x) \tan (x) + 2 \sec (x) \cos (2 x)$

Reorder terms.

$\sec (x) \sin (2 x) \tan (x) + 2 \cos (2 x) \sec (x)$

Find the Derivative – d/dx (sin(2x))/(cos(x))

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