Calculus Examples

$\frac{6 \cos (x)}{1 - \sin (x)}$

Since $6$ is constant with respect to $x$ , the derivative of $\frac{6 \cos (x)}{1 - \sin (x)}$ with respect to $x$ is $6 \frac{d}{d x} [\frac{\cos (x)}{1 - \sin (x)}]$ .

$6 \frac{d}{d x} [\frac{\cos (x)}{1 - \sin (x)}]$

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 $f (x) = \cos (x)$ and $g (x) = 1 - \sin (x)$ .

$6 \frac{(1 - \sin (x)) \frac{d}{d x} [\cos (x)] - \cos (x) \frac{d}{d x} [1 - \sin (x)]}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) - \cos (x) \frac{d}{d x} [1 - \sin (x)]}{{(1 - \sin (x))}^{2}}$

Differentiate.

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) - \cos (x) (\frac{d}{d x} [1] + \frac{d}{d x} [- \sin (x)])}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) - \cos (x) (0 + \frac{d}{d x} [- \sin (x)])}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) - \cos (x) \frac{d}{d x} [- \sin (x)]}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) - \cos (x) (- \frac{d}{d x} [\sin (x)])}{{(1 - \sin (x))}^{2}}$

Multiply.

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Multiply $- 1$ by $- 1$ .

$6 \frac{(1 - \sin (x)) (- \sin (x)) + 1 \cos (x) \frac{d}{d x} [\sin (x)]}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) + \cos (x) \frac{d}{d x} [\sin (x)]}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) + \cos (x) \cos (x)}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) + \cos^{1} (x) \cos (x)}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) + \cos^{1} (x) \cos^{1} (x)}{{(1 - \sin (x))}^{2}}$

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

$6 \frac{(1 - \sin (x)) (- \sin (x)) + {\cos (x)}^{1 + 1}}{{(1 - \sin (x))}^{2}}$

Add $1$ and $1$ .

$6 \frac{(1 - \sin (x)) (- \sin (x)) + \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

Combine $6$ and $\frac{(1 - \sin (x)) (- \sin (x)) + \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$ .

$\frac{6 ((1 - \sin (x)) (- \sin (x)) + \cos^{2} (x))}{{(1 - \sin (x))}^{2}}$

Simplify.

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Apply the distributive property.

$\frac{6 (1 (- \sin (x)) - \sin (x) (- \sin (x)) + \cos^{2} (x))}{{(1 - \sin (x))}^{2}}$

Apply the distributive property.

$\frac{6 (1 (- \sin (x))) + 6 (- \sin (x) (- \sin (x))) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

Simplify the numerator.

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Simplify each term.

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

$\frac{6 (- \sin (x)) + 6 (- \sin (x) (- \sin (x))) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

Multiply $- 1$ by $6$ .

$\frac{- 6 \sin (x) + 6 (- \sin (x) (- \sin (x))) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

Multiply $- \sin (x) (- \sin (x))$ .

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Multiply $- 1$ by $- 1$ .

$\frac{- 6 \sin (x) + 6 (1 \sin (x) \sin (x)) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

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

$\frac{- 6 \sin (x) + 6 (\sin (x) \sin (x)) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

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

$\frac{- 6 \sin (x) + 6 (\sin^{1} (x) \sin (x)) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

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

$\frac{- 6 \sin (x) + 6 (\sin^{1} (x) \sin^{1} (x)) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

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

$\frac{- 6 \sin (x) + 6 {\sin (x)}^{1 + 1} + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

Add $1$ and $1$ .

$\frac{- 6 \sin (x) + 6 \sin^{2} (x) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

Factor $6$ out of $6 \sin^{2} (x)$ .

$\frac{- 6 \sin (x) + 6 (\sin^{2} (x)) + 6 \cos^{2} (x)}{{(1 - \sin (x))}^{2}}$

Factor $6$ out of $6 \cos^{2} (x)$ .

$\frac{- 6 \sin (x) + 6 (\sin^{2} (x)) + 6 (\cos^{2} (x))}{{(1 - \sin (x))}^{2}}$

Factor $6$ out of $6 (\sin^{2} (x)) + 6 (\cos^{2} (x))$ .

$\frac{- 6 \sin (x) + 6 (\sin^{2} (x) + \cos^{2} (x))}{{(1 - \sin (x))}^{2}}$

Apply pythagorean identity.

$\frac{- 6 \sin (x) + 6 \cdot 1}{{(1 - \sin (x))}^{2}}$

Multiply $6$ by $1$ .

$\frac{- 6 \sin (x) + 6}{{(1 - \sin (x))}^{2}}$

Reorder terms.

$\frac{6 - 6 \sin (x)}{{(1 - \sin (x))}^{2}}$

Factor $6$ out of $6 - 6 \sin (x)$ .

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Factor $6$ out of $6$ .

$\frac{6 (1) - 6 \sin (x)}{{(1 - \sin (x))}^{2}}$

Factor $6$ out of $- 6 \sin (x)$ .

$\frac{6 (1) + 6 (- \sin (x))}{{(1 - \sin (x))}^{2}}$

Factor $6$ out of $6 (1) + 6 (- \sin (x))$ .

$\frac{6 (1 - \sin (x))}{{(1 - \sin (x))}^{2}}$

Cancel the common factor of $1 - \sin (x)$ and ${(1 - \sin (x))}^{2}$ .

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Factor $1 - \sin (x)$ out of $6 (1 - \sin (x))$ .

$\frac{(1 - \sin (x)) \cdot 6}{{(1 - \sin (x))}^{2}}$

Cancel the common factors.

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Factor $1 - \sin (x)$ out of ${(1 - \sin (x))}^{2}$ .

$\frac{(1 - \sin (x)) \cdot 6}{(1 - \sin (x)) (1 - \sin (x))}$

Cancel the common factor.

$\frac{(1 - \sin (x)) \cdot 6}{(1 - \sin (x)) (1 - \sin (x))}$

Rewrite the expression.

$\frac{6}{1 - \sin (x)}$