Calculus Examples

$y = \frac{1}{4 - 5 x - x^{2}}$

Step 1

Rewrite $\frac{1}{4 - 5 x - x^{2}}$ as ${(4 - 5 x - x^{2})}^{- 1}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 4 - 5 x - x^{2}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $4 - 5 x - x^{2}$ .

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

Step 2.2

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

$- u^{- 2} \frac{d}{d x} [4 - 5 x - x^{2}]$

Step 2.3

Replace all occurrences of $u$ with $4 - 5 x - x^{2}$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $4 - 5 x - x^{2}$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- x^{2}]$ .

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

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$- {(4 - 5 x - x^{2})}^{- 2} (0 + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [- x^{2}])$

Step 3.3

Add $0$ and $\frac{d}{d x} [- 5 x]$ .

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

Step 3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

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

Step 3.5

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

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

Step 3.6

Multiply $- 5$ by $1$ .

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

Step 3.7

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

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

Step 3.8

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

$- {(4 - 5 x - x^{2})}^{- 2} (- 5 - (2 x))$

Step 3.9

Multiply $2$ by $- 1$ .

$- {(4 - 5 x - x^{2})}^{- 2} (- 5 - 2 x)$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Reorder the factors of $- \frac{1}{{(4 - 5 x - x^{2})}^{2}} (- 5 - 2 x)$ .

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Multiply $- 1$ by $- 5$ .

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

Step 4.5

Multiply $- 2$ by $- 1$ .

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

Step 4.6

Multiply $5 + 2 x$ by $\frac{1}{{(4 - 5 x - x^{2})}^{2}}$ .

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