Calculus Examples

$y = \frac{1}{3 x - 5}$ , $x = 1$

Step 1

Rewrite $\frac{1}{3 x - 5}$ as ${(3 x - 5)}^{- 1}$ .

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

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 $g (x) = 3 x - 5$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $3 x - 5$ .

$\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [3 x - 5]$

Step 2.2

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

$- u^{- 2} \frac{d}{d x} [3 x - 5]$

Step 2.3

Replace all occurrences of $u$ with $3 x - 5$ .

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

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $3 x - 5$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 5]$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $3$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $3$ and $0$ .

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

Step 3.6.2

Multiply $3$ by $- 1$ .

$- 3 {(3 x - 5)}^{- 2}$

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Combine terms.

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Step 4.2.1

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

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

Step 4.2.2

Move the negative in front of the fraction.

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

Step 5

Evaluate the derivative at $x = 1$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Multiply $3$ by $1$ .

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

Step 6.2

Subtract $5$ from $3$ .

$- \frac{3}{{(- 2)}^{2}}$

Step 6.3

Raise $- 2$ to the power of $2$ .

$- \frac{3}{4}$