Calculus Examples

$f (x) = \frac{x}{{(7 x - 6)}^{8}}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x$ and $g (x) = {(7 x - 6)}^{8}$ .

$\frac{{(7 x - 6)}^{8} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 x - 6)}^{8}]}{{({(7 x - 6)}^{8})}^{2}}$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Multiply the exponents in ${({(7 x - 6)}^{8})}^{2}$ .

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Step 2.1.1

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

$\frac{{(7 x - 6)}^{8} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 x - 6)}^{8}]}{{(7 x - 6)}^{8 \cdot 2}}$

Step 2.1.2

Multiply $8$ by $2$ .

$\frac{{(7 x - 6)}^{8} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 x - 6)}^{8}]}{{(7 x - 6)}^{16}}$

Step 2.2

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

$\frac{{(7 x - 6)}^{8} \cdot 1 - x \frac{d}{d x} [{(7 x - 6)}^{8}]}{{(7 x - 6)}^{16}}$

Step 2.3

Multiply ${(7 x - 6)}^{8}$ by $1$ .

$\frac{{(7 x - 6)}^{8} - x \frac{d}{d x} [{(7 x - 6)}^{8}]}{{(7 x - 6)}^{16}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{8}$ and $g (x) = 7 x - 6$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $7 x - 6$ .

$\frac{{(7 x - 6)}^{8} - x (\frac{d}{d u} [u^{8}] \frac{d}{d x} [7 x - 6])}{{(7 x - 6)}^{16}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 8$ .

$\frac{{(7 x - 6)}^{8} - x (8 u^{7} \frac{d}{d x} [7 x - 6])}{{(7 x - 6)}^{16}}$

Step 3.3

Replace all occurrences of $u$ with $7 x - 6$ .

$\frac{{(7 x - 6)}^{8} - x (8 {(7 x - 6)}^{7} \frac{d}{d x} [7 x - 6])}{{(7 x - 6)}^{16}}$

Step 4

Simplify with factoring out.

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Step 4.1

Multiply $8$ by $- 1$ .

$\frac{{(7 x - 6)}^{8} - 8 x ({(7 x - 6)}^{7} \frac{d}{d x} [7 x - 6])}{{(7 x - 6)}^{16}}$

Step 4.2

Factor ${(7 x - 6)}^{7}$ out of ${(7 x - 6)}^{8} - 8 x ({(7 x - 6)}^{7} \frac{d}{d x} [7 x - 6])$ .

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Step 4.2.1

Factor ${(7 x - 6)}^{7}$ out of ${(7 x - 6)}^{8}$ .

$\frac{{(7 x - 6)}^{7} (7 x - 6) - 8 x ({(7 x - 6)}^{7} \frac{d}{d x} [7 x - 6])}{{(7 x - 6)}^{16}}$

Step 4.2.2

Factor ${(7 x - 6)}^{7}$ out of $- 8 x ({(7 x - 6)}^{7} \frac{d}{d x} [7 x - 6])$ .

$\frac{{(7 x - 6)}^{7} (7 x - 6) + {(7 x - 6)}^{7} (- 8 x (\frac{d}{d x} [7 x - 6]))}{{(7 x - 6)}^{16}}$

Step 4.2.3

Factor ${(7 x - 6)}^{7}$ out of ${(7 x - 6)}^{7} (7 x - 6) + {(7 x - 6)}^{7} (- 8 x (\frac{d}{d x} [7 x - 6]))$ .

$\frac{{(7 x - 6)}^{7} (7 x - 6 - 8 x (\frac{d}{d x} [7 x - 6]))}{{(7 x - 6)}^{16}}$

Step 5

Cancel the common factors.

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Step 5.1

Factor ${(7 x - 6)}^{7}$ out of ${(7 x - 6)}^{16}$ .

$\frac{{(7 x - 6)}^{7} (7 x - 6 - 8 x (\frac{d}{d x} [7 x - 6]))}{{(7 x - 6)}^{7} {(7 x - 6)}^{9}}$

Step 5.2

Cancel the common factor.

$\frac{{(7 x - 6)}^{7} (7 x - 6 - 8 x (\frac{d}{d x} [7 x - 6]))}{{(7 x - 6)}^{7} {(7 x - 6)}^{9}}$

Step 5.3

Rewrite the expression.

$\frac{7 x - 6 - 8 x (\frac{d}{d x} [7 x - 6])}{{(7 x - 6)}^{9}}$

Step 6

By the Sum Rule, the derivative of $7 x - 6$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 6]$ .

$\frac{7 x - 6 - 8 x (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 6])}{{(7 x - 6)}^{9}}$

Step 7

Since $7$ is constant with respect to $x$ , the derivative of $7 x$ with respect to $x$ is $7 \frac{d}{d x} [x]$ .

$\frac{7 x - 6 - 8 x (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 6])}{{(7 x - 6)}^{9}}$

Step 8

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

$\frac{7 x - 6 - 8 x (7 \cdot 1 + \frac{d}{d x} [- 6])}{{(7 x - 6)}^{9}}$

Step 9

Multiply $7$ by $1$ .

$\frac{7 x - 6 - 8 x (7 + \frac{d}{d x} [- 6])}{{(7 x - 6)}^{9}}$

Step 10

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{7 x - 6 - 8 x (7 + 0)}{{(7 x - 6)}^{9}}$

Step 11

Simplify by adding terms.

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Step 11.1

Add $7$ and $0$ .

$\frac{7 x - 6 - 8 x \cdot 7}{{(7 x - 6)}^{9}}$

Step 11.2

Multiply $7$ by $- 8$ .

$\frac{7 x - 6 - 56 x}{{(7 x - 6)}^{9}}$

Step 11.3

Subtract $56 x$ from $7 x$ .

$\frac{- 49 x - 6}{{(7 x - 6)}^{9}}$

Step 12

Simplify.

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Step 12.1

Factor $- 1$ out of $- 49 x$ .

$\frac{- (49 x) - 6}{{(7 x - 6)}^{9}}$

Step 12.2

Rewrite $- 6$ as $- 1 (6)$ .

$\frac{- (49 x) - 1 (6)}{{(7 x - 6)}^{9}}$

Step 12.3

Factor $- 1$ out of $- (49 x) - 1 (6)$ .

$\frac{- (49 x + 6)}{{(7 x - 6)}^{9}}$

Step 12.4

Rewrite $- (49 x + 6)$ as $- 1 (49 x + 6)$ .

$\frac{- 1 (49 x + 6)}{{(7 x - 6)}^{9}}$

Step 12.5

Move the negative in front of the fraction.

$- \frac{49 x + 6}{{(7 x - 6)}^{9}}$