Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

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 $g (x) = x \sin (x)$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x \sin (x)$ .

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

Step 3.1.2

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

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

Step 3.1.3

Replace all occurrences of $u$ with $x \sin (x)$ .

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

Step 3.2

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 $f (x) = x$ and $g (x) = \sin (x)$ .

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

Step 3.3

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

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

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

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

Step 3.4.2

Multiply $\sin (x)$ by $1$ .

$\sec^{2} (x \sin (x)) (x \cos (x) + \sin (x))$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$\sec^{2} (x \sin (x)) (x \cos (x)) + \sec^{2} (x \sin (x)) \sin (x)$

Step 3.5.2

Reorder terms.

$x \sec^{2} (x \sin (x)) \cos (x) + \sec^{2} (x \sin (x)) \sin (x)$

Step 4

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

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

Step 5

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

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