Calculus Examples

$\frac{x + 3}{{(x - 3)}^{5}}$

Step 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 $f (x) = x + 3$ and $g (x) = {(x - 3)}^{5}$ .

$\frac{{(x - 3)}^{5} \frac{d}{d x} [x + 3] - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{({(x - 3)}^{5})}^{2}}$

Step 2

Differentiate.

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Step 2.1

Multiply the exponents in ${({(x - 3)}^{5})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(x - 3)}^{5} \frac{d}{d x} [x + 3] - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{(x - 3)}^{5 \cdot 2}}$

Step 2.1.2

Multiply $5$ by $2$ .

$\frac{{(x - 3)}^{5} \frac{d}{d x} [x + 3] - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{(x - 3)}^{10}}$

Step 2.2

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{{(x - 3)}^{5} (\frac{d}{d x} [x] + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{(x - 3)}^{10}}$

Step 2.3

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

$\frac{{(x - 3)}^{5} (1 + \frac{d}{d x} [3]) - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{(x - 3)}^{10}}$

Step 2.4

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$\frac{{(x - 3)}^{5} (1 + 0) - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{(x - 3)}^{10}}$

Step 2.5

Simplify the expression.

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Step 2.5.1

Add $1$ and $0$ .

$\frac{{(x - 3)}^{5} \cdot 1 - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{(x - 3)}^{10}}$

Step 2.5.2

Multiply ${(x - 3)}^{5}$ by $1$ .

$\frac{{(x - 3)}^{5} - (x + 3) \frac{d}{d x} [{(x - 3)}^{5}]}{{(x - 3)}^{10}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x - 3$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x - 3$ .

$\frac{{(x - 3)}^{5} - (x + 3) (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x - 3])}{{(x - 3)}^{10}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$\frac{{(x - 3)}^{5} - (x + 3) (5 u^{4} \frac{d}{d x} [x - 3])}{{(x - 3)}^{10}}$

Step 3.3

Replace all occurrences of $u$ with $x - 3$ .

$\frac{{(x - 3)}^{5} - (x + 3) (5 {(x - 3)}^{4} \frac{d}{d x} [x - 3])}{{(x - 3)}^{10}}$

Step 4

Simplify with factoring out.

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Step 4.1

Multiply $5$ by $- 1$ .

$\frac{{(x - 3)}^{5} - 5 (x + 3) ({(x - 3)}^{4} \frac{d}{d x} [x - 3])}{{(x - 3)}^{10}}$

Step 4.2

Factor ${(x - 3)}^{4}$ out of ${(x - 3)}^{5} - 5 (x + 3) ({(x - 3)}^{4} \frac{d}{d x} [x - 3])$ .

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Step 4.2.1

Factor ${(x - 3)}^{4}$ out of ${(x - 3)}^{5}$ .

$\frac{{(x - 3)}^{4} (x - 3) - 5 (x + 3) ({(x - 3)}^{4} \frac{d}{d x} [x - 3])}{{(x - 3)}^{10}}$

Step 4.2.2

Factor ${(x - 3)}^{4}$ out of $- 5 (x + 3) ({(x - 3)}^{4} \frac{d}{d x} [x - 3])$ .

$\frac{{(x - 3)}^{4} (x - 3) + {(x - 3)}^{4} (- 5 (x + 3) (\frac{d}{d x} [x - 3]))}{{(x - 3)}^{10}}$

Step 4.2.3

Factor ${(x - 3)}^{4}$ out of ${(x - 3)}^{4} (x - 3) + {(x - 3)}^{4} (- 5 (x + 3) (\frac{d}{d x} [x - 3]))$ .

$\frac{{(x - 3)}^{4} (x - 3 - 5 (x + 3) (\frac{d}{d x} [x - 3]))}{{(x - 3)}^{10}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor ${(x - 3)}^{4}$ out of ${(x - 3)}^{10}$ .

$\frac{{(x - 3)}^{4} (x - 3 - 5 (x + 3) (\frac{d}{d x} [x - 3]))}{{(x - 3)}^{4} {(x - 3)}^{6}}$

Step 5.2

Cancel the common factor.

$\frac{{(x - 3)}^{4} (x - 3 - 5 (x + 3) (\frac{d}{d x} [x - 3]))}{{(x - 3)}^{4} {(x - 3)}^{6}}$

Step 5.3

Rewrite the expression.

$\frac{x - 3 - 5 (x + 3) (\frac{d}{d x} [x - 3])}{{(x - 3)}^{6}}$

Step 6

By the Sum Rule, the derivative of $x - 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 3]$ .

$\frac{x - 3 - 5 (x + 3) (\frac{d}{d x} [x] + \frac{d}{d x} [- 3])}{{(x - 3)}^{6}}$

Step 7

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

$\frac{x - 3 - 5 (x + 3) (1 + \frac{d}{d x} [- 3])}{{(x - 3)}^{6}}$

Step 8

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

$\frac{x - 3 - 5 (x + 3) (1 + 0)}{{(x - 3)}^{6}}$

Step 9

Simplify the expression.

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Step 9.1

Add $1$ and $0$ .

$\frac{x - 3 - 5 (x + 3) \cdot 1}{{(x - 3)}^{6}}$

Step 9.2

Multiply $- 5$ by $1$ .

$\frac{x - 3 - 5 (x + 3)}{{(x - 3)}^{6}}$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$\frac{x - 3 - 5 x - 5 \cdot 3}{{(x - 3)}^{6}}$

Step 10.2

Simplify the numerator.

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Step 10.2.1

Multiply $- 5$ by $3$ .

$\frac{x - 3 - 5 x - 15}{{(x - 3)}^{6}}$

Step 10.2.2

Subtract $5 x$ from $x$ .

$\frac{- 4 x - 3 - 15}{{(x - 3)}^{6}}$

Step 10.2.3

Subtract $15$ from $- 3$ .

$\frac{- 4 x - 18}{{(x - 3)}^{6}}$

Step 10.3

Factor $2$ out of $- 4 x - 18$ .

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Step 10.3.1

Factor $2$ out of $- 4 x$ .

$\frac{2 (- 2 x) - 18}{{(x - 3)}^{6}}$

Step 10.3.2

Factor $2$ out of $- 18$ .

$\frac{2 (- 2 x) + 2 (- 9)}{{(x - 3)}^{6}}$

Step 10.3.3

Factor $2$ out of $2 (- 2 x) + 2 (- 9)$ .

$\frac{2 (- 2 x - 9)}{{(x - 3)}^{6}}$

Step 10.4

Factor $- 1$ out of $- 2 x$ .

$\frac{2 (- (2 x) - 9)}{{(x - 3)}^{6}}$

Step 10.5

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{2 (- (2 x) - 1 (9))}{{(x - 3)}^{6}}$

Step 10.6

Factor $- 1$ out of $- (2 x) - 1 (9)$ .

$\frac{2 (- (2 x + 9))}{{(x - 3)}^{6}}$

Step 10.7

Rewrite $- (2 x + 9)$ as $- 1 (2 x + 9)$ .

$\frac{2 (- 1 (2 x + 9))}{{(x - 3)}^{6}}$

Step 10.8

Move the negative in front of the fraction.

$- \frac{2 (2 x + 9)}{{(x - 3)}^{6}}$