Calculus Examples

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

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 $g (x) = {(7 x - 9)}^{9}$ .

$\frac{{(7 x - 9)}^{9} \frac{d}{d x} [x] - x \frac{d}{d x} [{(7 x - 9)}^{9}]}{{({(7 x - 9)}^{9})}^{2}}$

Step 2

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

$\frac{{(7 x - 9)}^{9} \cdot 1 - x \frac{d}{d x} [{(7 x - 9)}^{9}]}{{({(7 x - 9)}^{9})}^{2}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{9}$ and $g (x) = 7 x - 9$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $7 x - 9$ .

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (\frac{d}{d u} [u^{9}] \frac{d}{d x} [7 x - 9])}{{({(7 x - 9)}^{9})}^{2}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 9$ .

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 u^{8} \frac{d}{d x} [7 x - 9])}{{({(7 x - 9)}^{9})}^{2}}$

Step 3.3

Replace all occurrences of $u$ with $7 x - 9$ .

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 {(7 x - 9)}^{8} \frac{d}{d x} [7 x - 9])}{{({(7 x - 9)}^{9})}^{2}}$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $7 x - 9$ with respect to $x$ is $\frac{d}{d x} [7 x] + \frac{d}{d x} [- 9]$ .

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 {(7 x - 9)}^{8} (\frac{d}{d x} [7 x] + \frac{d}{d x} [- 9]))}{{({(7 x - 9)}^{9})}^{2}}$

Step 4.2

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

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 {(7 x - 9)}^{8} (7 \frac{d}{d x} [x] + \frac{d}{d x} [- 9]))}{{({(7 x - 9)}^{9})}^{2}}$

Step 4.3

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

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 {(7 x - 9)}^{8} (7 \cdot 1 + \frac{d}{d x} [- 9]))}{{({(7 x - 9)}^{9})}^{2}}$

Step 4.4

Multiply $7$ by $1$ .

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 {(7 x - 9)}^{8} (7 + \frac{d}{d x} [- 9]))}{{({(7 x - 9)}^{9})}^{2}}$

Step 4.5

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

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 {(7 x - 9)}^{8} (7 + 0))}{{({(7 x - 9)}^{9})}^{2}}$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $7$ and $0$ .

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (9 {(7 x - 9)}^{8} \cdot 7)}{{({(7 x - 9)}^{9})}^{2}}$

Step 4.6.2

Multiply $7$ by $9$ .

$\frac{{(7 x - 9)}^{9} \cdot 1 - x (63 {(7 x - 9)}^{8})}{{({(7 x - 9)}^{9})}^{2}}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor ${(7 x - 9)}^{8}$ out of ${(7 x - 9)}^{9} \cdot 1 - x (63 {(7 x - 9)}^{8})$ .

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Step 5.1.1.1

Factor ${(7 x - 9)}^{8}$ out of ${(7 x - 9)}^{9} \cdot 1$ .

$\frac{{(7 x - 9)}^{8} ((7 x - 9) \cdot 1) - x (63 {(7 x - 9)}^{8})}{{({(7 x - 9)}^{9})}^{2}}$

Step 5.1.1.2

Factor ${(7 x - 9)}^{8}$ out of $- x (63 {(7 x - 9)}^{8})$ .

$\frac{{(7 x - 9)}^{8} ((7 x - 9) \cdot 1) + {(7 x - 9)}^{8} (- x \cdot (63))}{{({(7 x - 9)}^{9})}^{2}}$

Step 5.1.1.3

Factor ${(7 x - 9)}^{8}$ out of ${(7 x - 9)}^{8} ((7 x - 9) \cdot 1) + {(7 x - 9)}^{8} (- x \cdot (63))$ .

$\frac{{(7 x - 9)}^{8} ((7 x - 9) \cdot 1 - x \cdot (63))}{{({(7 x - 9)}^{9})}^{2}}$

Step 5.1.2

Multiply $7 x - 9$ by $1$ .

$\frac{{(7 x - 9)}^{8} (7 x - 9 - x \cdot (63))}{{({(7 x - 9)}^{9})}^{2}}$

Step 5.1.3

Multiply $63$ by $- 1$ .

$\frac{{(7 x - 9)}^{8} (7 x - 9 - 63 x)}{{({(7 x - 9)}^{9})}^{2}}$

Step 5.1.4

Subtract $63 x$ from $7 x$ .

$\frac{{(7 x - 9)}^{8} (- 56 x - 9)}{{({(7 x - 9)}^{9})}^{2}}$

Step 5.2

Combine terms.

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Step 5.2.1

Multiply the exponents in ${({(7 x - 9)}^{9})}^{2}$ .

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Step 5.2.1.1

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

$\frac{{(7 x - 9)}^{8} (- 56 x - 9)}{{(7 x - 9)}^{9 \cdot 2}}$

Step 5.2.1.2

Multiply $9$ by $2$ .

$\frac{{(7 x - 9)}^{8} (- 56 x - 9)}{{(7 x - 9)}^{18}}$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor ${(7 x - 9)}^{8}$ out of ${(7 x - 9)}^{18}$ .

$\frac{{(7 x - 9)}^{8} (- 56 x - 9)}{{(7 x - 9)}^{8} {(7 x - 9)}^{10}}$

Step 5.2.2.2

Cancel the common factor.

$\frac{{(7 x - 9)}^{8} (- 56 x - 9)}{{(7 x - 9)}^{8} {(7 x - 9)}^{10}}$

Step 5.2.2.3

Rewrite the expression.

$\frac{- 56 x - 9}{{(7 x - 9)}^{10}}$

Step 5.3

Factor $- 1$ out of $- 56 x$ .

$\frac{- (56 x) - 9}{{(7 x - 9)}^{10}}$

Step 5.4

Rewrite $- 9$ as $- 1 (9)$ .

$\frac{- (56 x) - 1 (9)}{{(7 x - 9)}^{10}}$

Step 5.5

Factor $- 1$ out of $- (56 x) - 1 (9)$ .

$\frac{- (56 x + 9)}{{(7 x - 9)}^{10}}$

Step 5.6

Rewrite $- (56 x + 9)$ as $- 1 (56 x + 9)$ .

$\frac{- 1 (56 x + 9)}{{(7 x - 9)}^{10}}$

Step 5.7

Move the negative in front of the fraction.

$- \frac{56 x + 9}{{(7 x - 9)}^{10}}$