Calculus Examples

$\frac{1}{x^{2} + 3 x - 1}$

Step 1

Rewrite $\frac{1}{x^{2} + 3 x - 1}$ as ${(x^{2} + 3 x - 1)}^{- 1}$ .

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

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 $g (x) = x^{2} + 3 x - 1$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{2} + 3 x - 1$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{2} + 3 x - 1]$

Step 2.2

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

$- u^{- 2} \frac{d}{d x} [x^{2} + 3 x - 1]$

Step 2.3

Replace all occurrences of $u$ with $x^{2} + 3 x - 1$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} + 3 x - 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3 x] + \frac{d}{d x} [- 1]$ .

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

Step 3.2

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

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

Step 3.3

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

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

Step 3.4

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

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

Step 3.5

Multiply $3$ by $1$ .

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

Step 3.6

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

$- {(x^{2} + 3 x - 1)}^{- 2} (2 x + 3 + 0)$

Step 3.7

Add $2 x + 3$ and $0$ .

$- {(x^{2} + 3 x - 1)}^{- 2} (2 x + 3)$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Reorder the factors of $- \frac{1}{{(x^{2} + 3 x - 1)}^{2}} (2 x + 3)$ .

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

Step 4.3

Apply the distributive property.

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

Step 4.4

Multiply $2$ by $- 1$ .

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

Step 4.5

Multiply $- 1$ by $3$ .

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

Step 4.6

Multiply $- 2 x - 3$ by $\frac{1}{{(x^{2} + 3 x - 1)}^{2}}$ .

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

Step 4.7

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

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

Step 4.8

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 4.9

Factor $- 1$ out of $- (2 x) - 1 (3)$ .

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

Step 4.10

Rewrite $- (2 x + 3)$ as $- 1 (2 x + 3)$ .

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

Step 4.11

Move the negative in front of the fraction.

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