Calculus Examples

$y = 4 x {(3 x - 9)}^{- 4}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (4 x {(3 x - 9)}^{- 4})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x {(3 x - 9)}^{- 4}$ with respect to $x$ is $4 \frac{d}{d x} [x {(3 x - 9)}^{- 4}]$ .

$4 \frac{d}{d x} [x {(3 x - 9)}^{- 4}]$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(3 x - 9)}^{- 4}$ .

$4 (x \frac{d}{d x} [{(3 x - 9)}^{- 4}] + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 4}$ and $g (x) = 3 x - 9$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $3 x - 9$ .

$4 (x (\frac{d}{d u} [u^{- 4}] \frac{d}{d x} [3 x - 9]) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 4$ .

$4 (x (- 4 u^{- 5} \frac{d}{d x} [3 x - 9]) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.3.3

Replace all occurrences of $u$ with $3 x - 9$ .

$4 (x (- 4 {(3 x - 9)}^{- 5} \frac{d}{d x} [3 x - 9]) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4

Differentiate.

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Step 3.4.1

By the Sum Rule, the derivative of $3 x - 9$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9]$ .

$4 (x (- 4 {(3 x - 9)}^{- 5} (\frac{d}{d x} [3 x] + \frac{d}{d x} [- 9])) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4.2

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

$4 (x (- 4 {(3 x - 9)}^{- 5} (3 \frac{d}{d x} [x] + \frac{d}{d x} [- 9])) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4.3

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

$4 (x (- 4 {(3 x - 9)}^{- 5} (3 \cdot 1 + \frac{d}{d x} [- 9])) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4.4

Multiply $3$ by $1$ .

$4 (x (- 4 {(3 x - 9)}^{- 5} (3 + \frac{d}{d x} [- 9])) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4.5

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$4 (x (- 4 {(3 x - 9)}^{- 5} (3 + 0)) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4.6

Simplify the expression.

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Step 3.4.6.1

Add $3$ and $0$ .

$4 (x (- 4 {(3 x - 9)}^{- 5} \cdot 3) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4.6.2

Multiply $3$ by $- 4$ .

$4 (x (- 12 {(3 x - 9)}^{- 5}) + {(3 x - 9)}^{- 4} \frac{d}{d x} [x])$

Step 3.4.7

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

$4 (x (- 12 {(3 x - 9)}^{- 5}) + {(3 x - 9)}^{- 4} \cdot 1)$

Step 3.4.8

Multiply ${(3 x - 9)}^{- 4}$ by $1$ .

$4 (x (- 12 {(3 x - 9)}^{- 5}) + {(3 x - 9)}^{- 4})$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$4 (x (- 12 {(3 x - 9)}^{- 5})) + 4 {(3 x - 9)}^{- 4}$

Step 3.5.2

Multiply $- 12$ by $4$ .

$- 48 x {(3 x - 9)}^{- 5} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3

Simplify each term.

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Step 3.5.3.1

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

$- 48 x \frac{1}{{(3 x - 9)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.2

Simplify the denominator.

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Step 3.5.3.2.1

Factor $3$ out of $3 x - 9$ .

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Step 3.5.3.2.1.1

Factor $3$ out of $3 x$ .

$- 48 x \frac{1}{{(3 (x) - 9)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.2.1.2

Factor $3$ out of $- 9$ .

$- 48 x \frac{1}{{(3 x + 3 \cdot - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.2.1.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$- 48 x \frac{1}{{(3 (x - 3))}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.2.2

Apply the product rule to $3 (x - 3)$ .

$- 48 x \frac{1}{3^{5} {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.2.3

Raise $3$ to the power of $5$ .

$- 48 x \frac{1}{243 {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.3

Cancel the common factor of $3$ .

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Step 3.5.3.3.1

Factor $3$ out of $- 48 x$ .

$3 (- 16 x) \frac{1}{243 {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.3.2

Factor $3$ out of $243 {(x - 3)}^{5}$ .

$3 (- 16 x) \frac{1}{3 (81 {(x - 3)}^{5})} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.3.3

Cancel the common factor.

$3 (- 16 x) \frac{1}{3 (81 {(x - 3)}^{5})} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.3.4

Rewrite the expression.

$- 16 x \frac{1}{81 {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.4

Combine $- 16$ and $\frac{1}{81 {(x - 3)}^{5}}$ .

$x \frac{- 16}{81 {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.5

Combine $x$ and $\frac{- 16}{81 {(x - 3)}^{5}}$ .

$\frac{x \cdot - 16}{81 {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.6

Move $- 16$ to the left of $x$ .

$\frac{- 16 \cdot x}{81 {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.7

Move the negative in front of the fraction.

$- \frac{16 \cdot x}{81 {(x - 3)}^{5}} + 4 {(3 x - 9)}^{- 4}$

Step 3.5.3.8

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

$- \frac{16 x}{81 {(x - 3)}^{5}} + 4 \frac{1}{{(3 x - 9)}^{4}}$

Step 3.5.3.9

Simplify the denominator.

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Step 3.5.3.9.1

Factor $3$ out of $3 x - 9$ .

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Step 3.5.3.9.1.1

Factor $3$ out of $3 x$ .

$- \frac{16 x}{81 {(x - 3)}^{5}} + 4 \frac{1}{{(3 (x) - 9)}^{4}}$

Step 3.5.3.9.1.2

Factor $3$ out of $- 9$ .

$- \frac{16 x}{81 {(x - 3)}^{5}} + 4 \frac{1}{{(3 x + 3 \cdot - 3)}^{4}}$

Step 3.5.3.9.1.3

Factor $3$ out of $3 x + 3 \cdot - 3$ .

$- \frac{16 x}{81 {(x - 3)}^{5}} + 4 \frac{1}{{(3 (x - 3))}^{4}}$

Step 3.5.3.9.2

Apply the product rule to $3 (x - 3)$ .

$- \frac{16 x}{81 {(x - 3)}^{5}} + 4 \frac{1}{3^{4} {(x - 3)}^{4}}$

Step 3.5.3.9.3

Raise $3$ to the power of $4$ .

$- \frac{16 x}{81 {(x - 3)}^{5}} + 4 \frac{1}{81 {(x - 3)}^{4}}$

Step 3.5.3.10

Combine $4$ and $\frac{1}{81 {(x - 3)}^{4}}$ .

$- \frac{16 x}{81 {(x - 3)}^{5}} + \frac{4}{81 {(x - 3)}^{4}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = - \frac{16 x}{81 {(x - 3)}^{5}} + \frac{4}{81 {(x - 3)}^{4}}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = - \frac{16 x}{81 {(x - 3)}^{5}} + \frac{4}{81 {(x - 3)}^{4}}$