Calculus Examples

$y = \frac{\sin (x)}{1 + \sin (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{\sin (x)}{1 + \sin (x)})$

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 Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \sin (x)$ and $g (x) = 1 + \sin (x)$ .

$\frac{(1 + \sin (x)) \frac{d}{d x} [\sin (x)] - \sin (x) \frac{d}{d x} [1 + \sin (x)]}{{(1 + \sin (x))}^{2}}$

Step 3.2

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

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

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $1 + \sin (x)$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\sin (x)]$ .

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

Step 3.3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(1 + \sin (x)) \cos (x) - \sin (x) (0 + \frac{d}{d x} [\sin (x)])}{{(1 + \sin (x))}^{2}}$

Step 3.3.3

Add $0$ and $\frac{d}{d x} [\sin (x)]$ .

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

Step 3.4

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

$\frac{(1 + \sin (x)) \cos (x) - \sin (x) \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$\frac{1 \cos (x) + \sin (x) \cos (x) - \sin (x) \cos (x)}{{(1 + \sin (x))}^{2}}$

Step 3.5.2

Simplify the numerator.

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Step 3.5.2.1

Combine the opposite terms in $1 \cos (x) + \sin (x) \cos (x) - \sin (x) \cos (x)$ .

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Step 3.5.2.1.1

Subtract $\sin (x) \cos (x)$ from $\sin (x) \cos (x)$ .

$\frac{1 \cos (x) + 0}{{(1 + \sin (x))}^{2}}$

Step 3.5.2.1.2

Add $1 \cos (x)$ and $0$ .

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

Step 3.5.2.2

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

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

Step 4

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

$y' = \frac{\cos (x)}{{(1 + \sin (x))}^{2}}$

Step 5

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

$\frac{d y}{d x} = \frac{\cos (x)}{{(1 + \sin (x))}^{2}}$