Calculus Examples

$y = \frac{4}{\sqrt{- x^{2} + 2 x - 5}}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- x^{2} + 2 x - 5}$ as ${(- x^{2} + 2 x - 5)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\frac{4}{{(- x^{2} + 2 x - 5)}^{\frac{1}{2}}}]$

Step 1.2

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{{(- x^{2} + 2 x - 5)}^{\frac{1}{2}}}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{{(- x^{2} + 2 x - 5)}^{\frac{1}{2}}}]$ .

$4 \frac{d}{d x} [\frac{1}{{(- x^{2} + 2 x - 5)}^{\frac{1}{2}}}]$

Step 1.3

Apply basic rules of exponents.

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Step 1.3.1

Rewrite $\frac{1}{{(- x^{2} + 2 x - 5)}^{\frac{1}{2}}}$ as ${({(- x^{2} + 2 x - 5)}^{\frac{1}{2}})}^{- 1}$ .

$4 \frac{d}{d x} [{({(- x^{2} + 2 x - 5)}^{\frac{1}{2}})}^{- 1}]$

Step 1.3.2

Multiply the exponents in ${({(- x^{2} + 2 x - 5)}^{\frac{1}{2}})}^{- 1}$ .

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Step 1.3.2.1

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

$4 \frac{d}{d x} [{(- x^{2} + 2 x - 5)}^{\frac{1}{2} \cdot - 1}]$

Step 1.3.2.2

Combine $\frac{1}{2}$ and $- 1$ .

$4 \frac{d}{d x} [{(- x^{2} + 2 x - 5)}^{\frac{- 1}{2}}]$

Step 1.3.2.3

Move the negative in front of the fraction.

$4 \frac{d}{d x} [{(- x^{2} + 2 x - 5)}^{- \frac{1}{2}}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{2}}$ and $g (x) = - x^{2} + 2 x - 5$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $- x^{2} + 2 x - 5$ .

$4 (\frac{d}{d u} [u^{- \frac{1}{2}}] \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{1}{2}$ .

$4 (- \frac{1}{2} u^{- \frac{1}{2} - 1} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 2.3

Replace all occurrences of $u$ with $- x^{2} + 2 x - 5$ .

$4 (- \frac{1}{2} {(- x^{2} + 2 x - 5)}^{- \frac{1}{2} - 1} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 (- \frac{1}{2} {(- x^{2} + 2 x - 5)}^{- \frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 4

Combine $- 1$ and $\frac{2}{2}$ .

$4 (- \frac{1}{2} {(- x^{2} + 2 x - 5)}^{- \frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 5

Combine the numerators over the common denominator.

$4 (- \frac{1}{2} {(- x^{2} + 2 x - 5)}^{\frac{- 1 - 1 \cdot 2}{2}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $2$ .

$4 (- \frac{1}{2} {(- x^{2} + 2 x - 5)}^{\frac{- 1 - 2}{2}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 6.2

Subtract $2$ from $- 1$ .

$4 (- \frac{1}{2} {(- x^{2} + 2 x - 5)}^{\frac{- 3}{2}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 7

Move the negative in front of the fraction.

$4 (- \frac{1}{2} {(- x^{2} + 2 x - 5)}^{- \frac{3}{2}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 8

Combine ${(- x^{2} + 2 x - 5)}^{- \frac{3}{2}}$ and $\frac{1}{2}$ .

$4 (- \frac{{(- x^{2} + 2 x - 5)}^{- \frac{3}{2}}}{2} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 9

Simplify the expression.

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Step 9.1

Move ${(- x^{2} + 2 x - 5)}^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$4 (- \frac{1}{2 {(- x^{2} + 2 x - 5)}^{\frac{3}{2}}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 9.2

Multiply $- 1$ by $4$ .

$- 4 (\frac{1}{2 {(- x^{2} + 2 x - 5)}^{\frac{3}{2}}} \frac{d}{d x} [- x^{2} + 2 x - 5])$

Step 10

Combine $\frac{1}{2 {(- x^{2} + 2 x - 5)}^{\frac{3}{2}}}$ and $- 4$ .

$\frac{- 4}{2 {(- x^{2} + 2 x - 5)}^{\frac{3}{2}}} \frac{d}{d x} [- x^{2} + 2 x - 5]$

Step 11

Factor $2$ out of $- 4$ .

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

Step 12

Cancel the common factors.

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Step 12.1

Factor $2$ out of $2 {(- x^{2} + 2 x - 5)}^{\frac{3}{2}}$ .

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Rewrite the expression.

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

Step 13

Move the negative in front of the fraction.

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

Step 14

By the Sum Rule, the derivative of $- x^{2} + 2 x - 5$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$ .

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

Step 15

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 16

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 17

Multiply $2$ by $- 1$ .

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

Step 18

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 19

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

$- \frac{2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}} (- 2 x + 2 \cdot 1 + \frac{d}{d x} [- 5])$

Step 20

Multiply $2$ by $1$ .

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

Step 21

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$- \frac{2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}} (- 2 x + 2 + 0)$

Step 22

Add $- 2 x + 2$ and $0$ .

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

Step 23

Simplify.

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Step 23.1

Reorder the factors of $- \frac{2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}} (- 2 x + 2)$ .

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

Step 23.2

Apply the distributive property.

$(- (- 2 x) - 1 \cdot 2) \frac{2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}}$

Step 23.3

Multiply $- 2$ by $- 1$ .

$(2 x - 1 \cdot 2) \frac{2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}}$

Step 23.4

Multiply $- 1$ by $2$ .

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

Step 23.5

Multiply $2 x - 2$ by $\frac{2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}}$ .

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

Step 23.6

Simplify the numerator.

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Step 23.6.1

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

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Step 23.6.1.1

Factor $2$ out of $2 x$ .

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

Step 23.6.1.2

Factor $2$ out of $- 2$ .

$\frac{(2 (x) + 2 (- 1)) \cdot 2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}}$

Step 23.6.1.3

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

$\frac{2 (x - 1) \cdot 2}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}}$

Step 23.6.2

Multiply $2$ by $2$ .

$\frac{4 (x - 1)}{{(- x^{2} + 2 x - 5)}^{\frac{3}{2}}}$