Calculus Examples

$f (x) = {(10 x + 4)}^{2} {(4 x^{2} - 7)}^{- 3}$

Step 1

Rewrite ${(10 x + 4)}^{2}$ as $(10 x + 4) (10 x + 4)$ .

$\frac{d}{d x} [(10 x + 4) (10 x + 4) {(4 x^{2} - 7)}^{- 3}]$

Step 2

Expand $(10 x + 4) (10 x + 4)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$\frac{d}{d x} [(10 x (10 x + 4) + 4 (10 x + 4)) {(4 x^{2} - 7)}^{- 3}]$

Step 2.2

Apply the distributive property.

$\frac{d}{d x} [(10 x (10 x) + 10 x \cdot 4 + 4 (10 x + 4)) {(4 x^{2} - 7)}^{- 3}]$

Step 2.3

Apply the distributive property.

$\frac{d}{d x} [(10 x (10 x) + 10 x \cdot 4 + 4 (10 x) + 4 \cdot 4) {(4 x^{2} - 7)}^{- 3}]$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [(10 \cdot 10 x \cdot x + 10 x \cdot 4 + 4 (10 x) + 4 \cdot 4) {(4 x^{2} - 7)}^{- 3}]$

Step 3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 3.1.2.1

Move $x$ .

$\frac{d}{d x} [(10 \cdot 10 (x \cdot x) + 10 x \cdot 4 + 4 (10 x) + 4 \cdot 4) {(4 x^{2} - 7)}^{- 3}]$

Step 3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [(10 \cdot 10 x^{2} + 10 x \cdot 4 + 4 (10 x) + 4 \cdot 4) {(4 x^{2} - 7)}^{- 3}]$

Step 3.1.3

Multiply $10$ by $10$ .

$\frac{d}{d x} [(100 x^{2} + 10 x \cdot 4 + 4 (10 x) + 4 \cdot 4) {(4 x^{2} - 7)}^{- 3}]$

Step 3.1.4

Multiply $4$ by $10$ .

$\frac{d}{d x} [(100 x^{2} + 40 x + 4 (10 x) + 4 \cdot 4) {(4 x^{2} - 7)}^{- 3}]$

Step 3.1.5

Multiply $10$ by $4$ .

$\frac{d}{d x} [(100 x^{2} + 40 x + 40 x + 4 \cdot 4) {(4 x^{2} - 7)}^{- 3}]$

Step 3.1.6

Multiply $4$ by $4$ .

$\frac{d}{d x} [(100 x^{2} + 40 x + 40 x + 16) {(4 x^{2} - 7)}^{- 3}]$

Step 3.2

Add $40 x$ and $40 x$ .

$\frac{d}{d x} [(100 x^{2} + 80 x + 16) {(4 x^{2} - 7)}^{- 3}]$

Step 4

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 $f (x) = 100 x^{2} + 80 x + 16$ and $g (x) = {(4 x^{2} - 7)}^{- 3}$ .

$(100 x^{2} + 80 x + 16) \frac{d}{d x} [{(4 x^{2} - 7)}^{- 3}] + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 5

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = 4 x^{2} - 7$ .

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Step 5.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 7$ .

$(100 x^{2} + 80 x + 16) (\frac{d}{d u} [u^{- 3}] \frac{d}{d x} [4 x^{2} - 7]) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 5.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 3$ .

$(100 x^{2} + 80 x + 16) (- 3 u^{- 4} \frac{d}{d x} [4 x^{2} - 7]) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 5.3

Replace all occurrences of $u$ with $4 x^{2} - 7$ .

$(100 x^{2} + 80 x + 16) (- 3 {(4 x^{2} - 7)}^{- 4} \frac{d}{d x} [4 x^{2} - 7]) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6

Differentiate.

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Step 6.1

By the Sum Rule, the derivative of $4 x^{2} - 7$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 7]$ .

$(100 x^{2} + 80 x + 16) (- 3 {(4 x^{2} - 7)}^{- 4} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 7])) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$(100 x^{2} + 80 x + 16) (- 3 {(4 x^{2} - 7)}^{- 4} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 7])) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6.3

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

$(100 x^{2} + 80 x + 16) (- 3 {(4 x^{2} - 7)}^{- 4} (4 (2 x) + \frac{d}{d x} [- 7])) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6.4

Multiply $2$ by $4$ .

$(100 x^{2} + 80 x + 16) (- 3 {(4 x^{2} - 7)}^{- 4} (8 x + \frac{d}{d x} [- 7])) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$(100 x^{2} + 80 x + 16) (- 3 {(4 x^{2} - 7)}^{- 4} (8 x + 0)) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6.6

Simplify the expression.

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Step 6.6.1

Add $8 x$ and $0$ .

$(100 x^{2} + 80 x + 16) (- 3 {(4 x^{2} - 7)}^{- 4} (8 x)) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6.6.2

Multiply $8$ by $- 3$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} \frac{d}{d x} [100 x^{2} + 80 x + 16]$

Step 6.7

By the Sum Rule, the derivative of $100 x^{2} + 80 x + 16$ with respect to $x$ is $\frac{d}{d x} [100 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [16]$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (\frac{d}{d x} [100 x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [16])$

Step 6.8

Since $100$ is constant with respect to $x$ , the derivative of $100 x^{2}$ with respect to $x$ is $100 \frac{d}{d x} [x^{2}]$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (100 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [80 x] + \frac{d}{d x} [16])$

Step 6.9

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

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (100 (2 x) + \frac{d}{d x} [80 x] + \frac{d}{d x} [16])$

Step 6.10

Multiply $2$ by $100$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (200 x + \frac{d}{d x} [80 x] + \frac{d}{d x} [16])$

Step 6.11

Since $80$ is constant with respect to $x$ , the derivative of $80 x$ with respect to $x$ is $80 \frac{d}{d x} [x]$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (200 x + 80 \frac{d}{d x} [x] + \frac{d}{d x} [16])$

Step 6.12

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

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (200 x + 80 \cdot 1 + \frac{d}{d x} [16])$

Step 6.13

Multiply $80$ by $1$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (200 x + 80 + \frac{d}{d x} [16])$

Step 6.14

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (200 x + 80 + 0)$

Step 6.15

Add $200 x + 80$ and $0$ .

$(100 x^{2} + 80 x + 16) (- 24 {(4 x^{2} - 7)}^{- 4} x) + {(4 x^{2} - 7)}^{- 3} (200 x + 80)$

Step 7

Simplify.

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Step 7.1

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

$(100 x^{2} + 80 x + 16) (- 24 \frac{1}{{(4 x^{2} - 7)}^{4}} x) + {(4 x^{2} - 7)}^{- 3} (200 x + 80)$

Step 7.2

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

$(100 x^{2} + 80 x + 16) (- 24 \frac{1}{{(4 x^{2} - 7)}^{4}} x) + \frac{1}{{(4 x^{2} - 7)}^{3}} (200 x + 80)$

Step 7.3

Combine terms.

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Step 7.3.1

Combine $- 24$ and $\frac{1}{{(4 x^{2} - 7)}^{4}}$ .

$(100 x^{2} + 80 x + 16) (\frac{- 24}{{(4 x^{2} - 7)}^{4}} x) + \frac{1}{{(4 x^{2} - 7)}^{3}} (200 x + 80)$

Step 7.3.2

Move the negative in front of the fraction.

$(100 x^{2} + 80 x + 16) (- \frac{24}{{(4 x^{2} - 7)}^{4}} x) + \frac{1}{{(4 x^{2} - 7)}^{3}} (200 x + 80)$

Step 7.3.3

Combine $x$ and $\frac{24}{{(4 x^{2} - 7)}^{4}}$ .

$(100 x^{2} + 80 x + 16) (- \frac{x \cdot 24}{{(4 x^{2} - 7)}^{4}}) + \frac{1}{{(4 x^{2} - 7)}^{3}} (200 x + 80)$

Step 7.3.4

Move $24$ to the left of $x$ .

$(100 x^{2} + 80 x + 16) (- \frac{24 x}{{(4 x^{2} - 7)}^{4}}) + \frac{1}{{(4 x^{2} - 7)}^{3}} (200 x + 80)$

Step 7.4

Reorder terms.

$- (100 x^{2} + 80 x + 16) \frac{24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5

Simplify each term.

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Step 7.5.1

Apply the distributive property.

$(- (100 x^{2}) - (80 x) - 1 \cdot 16) \frac{24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.2

Simplify.

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Step 7.5.2.1

Multiply $100$ by $- 1$ .

$(- 100 x^{2} - (80 x) - 1 \cdot 16) \frac{24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.2.2

Multiply $80$ by $- 1$ .

$(- 100 x^{2} - 80 x - 1 \cdot 16) \frac{24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.2.3

Multiply $- 1$ by $16$ .

$(- 100 x^{2} - 80 x - 16) \frac{24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.3

Multiply $- 100 x^{2} - 80 x - 16$ by $\frac{24 x}{{(4 x^{2} - 7)}^{4}}$ .

$\frac{(- 100 x^{2} - 80 x - 16) (24 x)}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4

Simplify the numerator.

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Step 7.5.4.1

Factor $4$ out of $- 100 x^{2} - 80 x - 16$ .

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Step 7.5.4.1.1

Factor $4$ out of $- 100 x^{2}$ .

$\frac{(4 (- 25 x^{2}) - 80 x - 16) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.1.2

Factor $4$ out of $- 80 x$ .

$\frac{(4 (- 25 x^{2}) + 4 (- 20 x) - 16) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.1.3

Factor $4$ out of $- 16$ .

$\frac{(4 (- 25 x^{2}) + 4 (- 20 x) + 4 (- 4)) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.1.4

Factor $4$ out of $4 (- 25 x^{2}) + 4 (- 20 x)$ .

$\frac{(4 (- 25 x^{2} - 20 x) + 4 (- 4)) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.1.5

Factor $4$ out of $4 (- 25 x^{2} - 20 x) + 4 (- 4)$ .

$\frac{4 (- 25 x^{2} - 20 x - 4) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.2

Factor by grouping.

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Step 7.5.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 25 \cdot - 4 = 100$ and whose sum is $b = - 20$ .

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Step 7.5.4.2.1.1

Factor $- 20$ out of $- 20 x$ .

$\frac{4 (- 25 x^{2} - 20 (x) - 4) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.2.1.2

Rewrite $- 20$ as $- 10$ plus $- 10$

$\frac{4 (- 25 x^{2} + (- 10 - 10) x - 4) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.2.1.3

Apply the distributive property.

$\frac{4 (- 25 x^{2} - 10 x - 10 x - 4) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.2.2

Factor out the greatest common factor from each group.

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Step 7.5.4.2.2.1

Group the first two terms and the last two terms.

$\frac{4 ((- 25 x^{2} - 10 x) - 10 x - 4) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 (5 x (- 5 x - 2) + 2 (- 5 x - 2)) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.2.3

Factor the polynomial by factoring out the greatest common factor, $- 5 x - 2$ .

$\frac{4 ((- 5 x - 2) (5 x + 2)) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

$\frac{4 (- 5 x - 2) (5 x + 2) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3

Combine exponents.

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Step 7.5.4.3.1

Factor $- 1$ out of $- 5 x$ .

$\frac{4 (- (5 x) - 2) (5 x + 2) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{4 (- (5 x) - 1 (2)) (5 x + 2) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.3

Factor $- 1$ out of $- (5 x) - 1 (2)$ .

$\frac{4 (- (5 x + 2)) (5 x + 2) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.4

Rewrite $- (5 x + 2)$ as $- 1 (5 x + 2)$ .

$\frac{4 (- 1 (5 x + 2)) (5 x + 2) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.5

Raise $5 x + 2$ to the power of $1$ .

$\frac{4 \cdot - 1 ({(5 x + 2)}^{1} (5 x + 2)) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.6

Raise $5 x + 2$ to the power of $1$ .

$\frac{4 \cdot - 1 ({(5 x + 2)}^{1} {(5 x + 2)}^{1}) \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{4 \cdot - 1 {(5 x + 2)}^{1 + 1} \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.8

Add $1$ and $1$ .

$\frac{4 \cdot - 1 {(5 x + 2)}^{2} \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.3.9

Multiply $4$ by $- 1$ .

$\frac{- 4 {(5 x + 2)}^{2} \cdot 24 x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.4.4

Multiply $24$ by $- 4$ .

$\frac{- 96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.5

Move the negative in front of the fraction.

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + (200 x + 80) \frac{1}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.6

Multiply $200 x + 80$ by $\frac{1}{{(4 x^{2} - 7)}^{3}}$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{200 x + 80}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.7

Factor $40$ out of $200 x + 80$ .

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Step 7.5.7.1

Factor $40$ out of $200 x$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x) + 80}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.7.2

Factor $40$ out of $80$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x) + 40 (2)}{{(4 x^{2} - 7)}^{3}}$

Step 7.5.7.3

Factor $40$ out of $40 (5 x) + 40 (2)$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x + 2)}{{(4 x^{2} - 7)}^{3}}$

Step 7.6

To write $\frac{40 (5 x + 2)}{{(4 x^{2} - 7)}^{3}}$ as a fraction with a common denominator, multiply by $\frac{4 x^{2} - 7}{4 x^{2} - 7}$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x + 2)}{{(4 x^{2} - 7)}^{3}} \cdot \frac{4 x^{2} - 7}{4 x^{2} - 7}$

Step 7.7

Write each expression with a common denominator of ${(4 x^{2} - 7)}^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.7.1

Multiply $\frac{40 (5 x + 2)}{{(4 x^{2} - 7)}^{3}}$ by $\frac{4 x^{2} - 7}{4 x^{2} - 7}$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x + 2) (4 x^{2} - 7)}{{(4 x^{2} - 7)}^{3} (4 x^{2} - 7)}$

Step 7.7.2

Multiply ${(4 x^{2} - 7)}^{3}$ by $4 x^{2} - 7$ by adding the exponents.

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Step 7.7.2.1

Multiply ${(4 x^{2} - 7)}^{3}$ by $4 x^{2} - 7$ .

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Step 7.7.2.1.1

Raise $4 x^{2} - 7$ to the power of $1$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x + 2) (4 x^{2} - 7)}{{(4 x^{2} - 7)}^{3} {(4 x^{2} - 7)}^{1}}$

Step 7.7.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x + 2) (4 x^{2} - 7)}{{(4 x^{2} - 7)}^{3 + 1}}$

Step 7.7.2.2

Add $3$ and $1$ .

$- \frac{96 {(5 x + 2)}^{2} x}{{(4 x^{2} - 7)}^{4}} + \frac{40 (5 x + 2) (4 x^{2} - 7)}{{(4 x^{2} - 7)}^{4}}$

Step 7.8

Combine the numerators over the common denominator.

$\frac{- 96 {(5 x + 2)}^{2} x + 40 (5 x + 2) (4 x^{2} - 7)}{{(4 x^{2} - 7)}^{4}}$

Step 7.9

Simplify the numerator.

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Step 7.9.1

Factor $8 (5 x + 2)$ out of $- 96 {(5 x + 2)}^{2} x + 40 (5 x + 2) (4 x^{2} - 7)$ .

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Step 7.9.1.1

Factor $8 (5 x + 2)$ out of $- 96 {(5 x + 2)}^{2} x$ .

$\frac{8 (5 x + 2) (- 12 (5 x + 2) x) + 40 (5 x + 2) (4 x^{2} - 7)}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.1.2

Factor $8 (5 x + 2)$ out of $40 (5 x + 2) (4 x^{2} - 7)$ .

$\frac{8 (5 x + 2) (- 12 (5 x + 2) x) + 8 (5 x + 2) (5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.1.3

Factor $8 (5 x + 2)$ out of $8 (5 x + 2) (- 12 (5 x + 2) x) + 8 (5 x + 2) (5 (4 x^{2} - 7))$ .

$\frac{8 (5 x + 2) (- 12 (5 x + 2) x + 5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.2

Apply the distributive property.

$\frac{8 (5 x + 2) ((- 12 (5 x) - 12 \cdot 2) x + 5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.3

Multiply $5$ by $- 12$ .

$\frac{8 (5 x + 2) ((- 60 x - 12 \cdot 2) x + 5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.4

Multiply $- 12$ by $2$ .

$\frac{8 (5 x + 2) ((- 60 x - 24) x + 5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.5

Apply the distributive property.

$\frac{8 (5 x + 2) (- 60 x \cdot x - 24 x + 5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.6

Multiply $x$ by $x$ by adding the exponents.

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Step 7.9.6.1

Move $x$ .

$\frac{8 (5 x + 2) (- 60 (x \cdot x) - 24 x + 5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.6.2

Multiply $x$ by $x$ .

$\frac{8 (5 x + 2) (- 60 x^{2} - 24 x + 5 (4 x^{2} - 7))}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.7

Apply the distributive property.

$\frac{8 (5 x + 2) (- 60 x^{2} - 24 x + 5 (4 x^{2}) + 5 \cdot - 7)}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.8

Multiply $4$ by $5$ .

$\frac{8 (5 x + 2) (- 60 x^{2} - 24 x + 20 x^{2} + 5 \cdot - 7)}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.9

Multiply $5$ by $- 7$ .

$\frac{8 (5 x + 2) (- 60 x^{2} - 24 x + 20 x^{2} - 35)}{{(4 x^{2} - 7)}^{4}}$

Step 7.9.10

Add $- 60 x^{2}$ and $20 x^{2}$ .

$\frac{8 (5 x + 2) (- 40 x^{2} - 24 x - 35)}{{(4 x^{2} - 7)}^{4}}$

Step 7.10

Factor $- 1$ out of $- 40 x^{2}$ .

$\frac{8 (5 x + 2) (- (40 x^{2}) - 24 x - 35)}{{(4 x^{2} - 7)}^{4}}$

Step 7.11

Factor $- 1$ out of $- 24 x$ .

$\frac{8 (5 x + 2) (- (40 x^{2}) - (24 x) - 35)}{{(4 x^{2} - 7)}^{4}}$

Step 7.12

Factor $- 1$ out of $- (40 x^{2}) - (24 x)$ .

$\frac{8 (5 x + 2) (- (40 x^{2} + 24 x) - 35)}{{(4 x^{2} - 7)}^{4}}$

Step 7.13

Rewrite $- 35$ as $- 1 (35)$ .

$\frac{8 (5 x + 2) (- (40 x^{2} + 24 x) - 1 (35))}{{(4 x^{2} - 7)}^{4}}$

Step 7.14

Factor $- 1$ out of $- (40 x^{2} + 24 x) - 1 (35)$ .

$\frac{8 (5 x + 2) (- (40 x^{2} + 24 x + 35))}{{(4 x^{2} - 7)}^{4}}$

Step 7.15

Rewrite $- (40 x^{2} + 24 x + 35)$ as $- 1 (40 x^{2} + 24 x + 35)$ .

$\frac{8 (5 x + 2) (- 1 (40 x^{2} + 24 x + 35))}{{(4 x^{2} - 7)}^{4}}$

Step 7.16

Move the negative in front of the fraction.

$- \frac{(8 (5 x + 2)) (40 x^{2} + 24 x + 35)}{{(4 x^{2} - 7)}^{4}}$

Step 7.17

Reorder factors in $- \frac{(8 (5 x + 2)) (40 x^{2} + 24 x + 35)}{{(4 x^{2} - 7)}^{4}}$ .

$- \frac{8 (5 x + 2) (40 x^{2} + 24 x + 35)}{{(4 x^{2} - 7)}^{4}}$