Calculus Examples

$y = \frac{3}{\sin (x) + \cos (x)}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{3}{\sin (x) + \cos (x)})$

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 Constant Multiple Rule.

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Step 3.1.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{\sin (x) + \cos (x)}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{\sin (x) + \cos (x)}]$ .

$3 \frac{d}{d x} [\frac{1}{\sin (x) + \cos (x)}]$

Step 3.1.2

Rewrite $\frac{1}{\sin (x) + \cos (x)}$ as ${(\sin (x) + \cos (x))}^{- 1}$ .

$3 \frac{d}{d x} [{(\sin (x) + \cos (x))}^{- 1}]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = \sin (x) + \cos (x)$ .

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Step 3.2.1

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

$3 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 3.2.2

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

$3 (- u^{- 2} \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 3.2.3

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

$3 (- {(\sin (x) + \cos (x))}^{- 2} \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 3.3

Differentiate using the Sum Rule.

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Step 3.3.1

Multiply $- 1$ by $3$ .

$- 3 ({(\sin (x) + \cos (x))}^{- 2} \frac{d}{d x} [\sin (x) + \cos (x)])$

Step 3.3.2

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

$- 3 {(\sin (x) + \cos (x))}^{- 2} (\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)])$

Step 3.4

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

$- 3 {(\sin (x) + \cos (x))}^{- 2} (\cos (x) + \frac{d}{d x} [\cos (x)])$

Step 3.5

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

$- 3 {(\sin (x) + \cos (x))}^{- 2} (\cos (x) - \sin (x))$

Step 3.6

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 3 \frac{1}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$

Step 3.7

Simplify.

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Step 3.7.1

Combine terms.

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Step 3.7.1.1

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

$\frac{- 3}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$

Step 3.7.1.2

Move the negative in front of the fraction.

$- \frac{3}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$

Step 3.7.2

Reorder the factors of $- \frac{3}{{(\sin (x) + \cos (x))}^{2}} (\cos (x) - \sin (x))$ .

$- (\cos (x) - \sin (x)) \frac{3}{{(\sin (x) + \cos (x))}^{2}}$

Step 4

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

$y' = - (\cos (x) - \sin (x)) \frac{3}{{(\sin (x) + \cos (x))}^{2}}$

Step 5

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

$\frac{d y}{d x} = - (\cos (x) - \sin (x)) \frac{3}{{(\sin (x) + \cos (x))}^{2}}$