Calculus Examples

$y = {(2 x - 5)}^{4} {(8 x^{2} - 5)}^{- 3}$

Step 1

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 $f (x) = {(2 x - 5)}^{4}$ and $g (x) = {(8 x^{2} - 5)}^{- 3}$ .

${(2 x - 5)}^{4} \frac{d}{d x} [{(8 x^{2} - 5)}^{- 3}] + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 3}$ and $g (x) = 8 x^{2} - 5$ .

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Step 2.1

To apply the Chain Rule, set $u_{1}$ as $8 x^{2} - 5$ .

${(2 x - 5)}^{4} (\frac{d}{d u_{1}} [{u_{1}}^{- 3}] \frac{d}{d x} [8 x^{2} - 5]) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 2.2

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

${(2 x - 5)}^{4} (- 3 {u_{1}}^{- 4} \frac{d}{d x} [8 x^{2} - 5]) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 2.3

Replace all occurrences of $u_{1}$ with $8 x^{2} - 5$ .

${(2 x - 5)}^{4} (- 3 {(8 x^{2} - 5)}^{- 4} \frac{d}{d x} [8 x^{2} - 5]) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $8 x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- 5]$ .

${(2 x - 5)}^{4} (- 3 {(8 x^{2} - 5)}^{- 4} (\frac{d}{d x} [8 x^{2}] + \frac{d}{d x} [- 5])) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{2}$ with respect to $x$ is $8 \frac{d}{d x} [x^{2}]$ .

${(2 x - 5)}^{4} (- 3 {(8 x^{2} - 5)}^{- 4} (8 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5])) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3.3

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

${(2 x - 5)}^{4} (- 3 {(8 x^{2} - 5)}^{- 4} (8 (2 x) + \frac{d}{d x} [- 5])) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3.4

Multiply $2$ by $8$ .

${(2 x - 5)}^{4} (- 3 {(8 x^{2} - 5)}^{- 4} (16 x + \frac{d}{d x} [- 5])) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3.5

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

${(2 x - 5)}^{4} (- 3 {(8 x^{2} - 5)}^{- 4} (16 x + 0)) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $16 x$ and $0$ .

${(2 x - 5)}^{4} (- 3 {(8 x^{2} - 5)}^{- 4} (16 x)) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 3.6.2

Multiply $16$ by $- 3$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + {(8 x^{2} - 5)}^{- 3} \frac{d}{d x} [{(2 x - 5)}^{4}]$

Step 4

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 $g (x) = 2 x - 5$ .

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Step 4.1

To apply the Chain Rule, set $u_{2}$ as $2 x - 5$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + {(8 x^{2} - 5)}^{- 3} (\frac{d}{d u_{2}} [{u_{2}}^{4}] \frac{d}{d x} [2 x - 5])$

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 4$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + {(8 x^{2} - 5)}^{- 3} (4 {u_{2}}^{3} \frac{d}{d x} [2 x - 5])$

Step 4.3

Replace all occurrences of $u_{2}$ with $2 x - 5$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + {(8 x^{2} - 5)}^{- 3} (4 {(2 x - 5)}^{3} \frac{d}{d x} [2 x - 5])$

Step 5

Differentiate.

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Step 5.1

Move $4$ to the left of ${(8 x^{2} - 5)}^{- 3}$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 4 \cdot {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3} \frac{d}{d x} [2 x - 5]$

Step 5.2

By the Sum Rule, the derivative of $2 x - 5$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 4 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5])$

Step 5.3

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

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 4 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5])$

Step 5.4

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

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 4 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3} (2 \cdot 1 + \frac{d}{d x} [- 5])$

Step 5.5

Multiply $2$ by $1$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 4 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3} (2 + \frac{d}{d x} [- 5])$

Step 5.6

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

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 4 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3} (2 + 0)$

Step 5.7

Simplify the expression.

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Step 5.7.1

Add $2$ and $0$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 4 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3} \cdot 2$

Step 5.7.2

Multiply $2$ by $4$ .

${(2 x - 5)}^{4} (- 48 {(8 x^{2} - 5)}^{- 4} x) + 8 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3}$

Step 6

Simplify.

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Step 6.1

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

${(2 x - 5)}^{4} (- 48 \frac{1}{{(8 x^{2} - 5)}^{4}} x) + 8 {(8 x^{2} - 5)}^{- 3} {(2 x - 5)}^{3}$

Step 6.2

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

${(2 x - 5)}^{4} (- 48 \frac{1}{{(8 x^{2} - 5)}^{4}} x) + 8 \frac{1}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3

Combine terms.

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Step 6.3.1

Combine $- 48$ and $\frac{1}{{(8 x^{2} - 5)}^{4}}$ .

${(2 x - 5)}^{4} (\frac{- 48}{{(8 x^{2} - 5)}^{4}} x) + 8 \frac{1}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3.2

Move the negative in front of the fraction.

${(2 x - 5)}^{4} (- \frac{48}{{(8 x^{2} - 5)}^{4}} x) + 8 \frac{1}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3.3

Combine $x$ and $\frac{48}{{(8 x^{2} - 5)}^{4}}$ .

${(2 x - 5)}^{4} (- \frac{x \cdot 48}{{(8 x^{2} - 5)}^{4}}) + 8 \frac{1}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3.4

Move $48$ to the left of $x$ .

${(2 x - 5)}^{4} (- \frac{48 \cdot x}{{(8 x^{2} - 5)}^{4}}) + 8 \frac{1}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3.5

Combine ${(2 x - 5)}^{4}$ and $\frac{48 x}{{(8 x^{2} - 5)}^{4}}$ .

$- \frac{{(2 x - 5)}^{4} (48 x)}{{(8 x^{2} - 5)}^{4}} + 8 \frac{1}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3.6

Move $48$ to the left of ${(2 x - 5)}^{4}$ .

$- \frac{48 \cdot {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + 8 \frac{1}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3.7

Combine $8$ and $\frac{1}{{(8 x^{2} - 5)}^{3}}$ .

$- \frac{48 {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + \frac{8}{{(8 x^{2} - 5)}^{3}} {(2 x - 5)}^{3}$

Step 6.3.8

Combine $\frac{8}{{(8 x^{2} - 5)}^{3}}$ and ${(2 x - 5)}^{3}$ .

$- \frac{48 {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + \frac{8 {(2 x - 5)}^{3}}{{(8 x^{2} - 5)}^{3}}$

Step 6.3.9

To write $\frac{8 {(2 x - 5)}^{3}}{{(8 x^{2} - 5)}^{3}}$ as a fraction with a common denominator, multiply by $\frac{8 x^{2} - 5}{8 x^{2} - 5}$ .

$- \frac{48 {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + \frac{8 {(2 x - 5)}^{3}}{{(8 x^{2} - 5)}^{3}} \cdot \frac{8 x^{2} - 5}{8 x^{2} - 5}$

Step 6.3.10

Write each expression with a common denominator of ${(8 x^{2} - 5)}^{4}$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.3.10.1

Multiply $\frac{8 {(2 x - 5)}^{3}}{{(8 x^{2} - 5)}^{3}}$ by $\frac{8 x^{2} - 5}{8 x^{2} - 5}$ .

$- \frac{48 {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + \frac{8 {(2 x - 5)}^{3} (8 x^{2} - 5)}{{(8 x^{2} - 5)}^{3} (8 x^{2} - 5)}$

Step 6.3.10.2

Multiply ${(8 x^{2} - 5)}^{3}$ by $8 x^{2} - 5$ by adding the exponents.

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Step 6.3.10.2.1

Multiply ${(8 x^{2} - 5)}^{3}$ by $8 x^{2} - 5$ .

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Step 6.3.10.2.1.1

Raise $8 x^{2} - 5$ to the power of $1$ .

$- \frac{48 {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + \frac{8 {(2 x - 5)}^{3} (8 x^{2} - 5)}{{(8 x^{2} - 5)}^{3} {(8 x^{2} - 5)}^{1}}$

Step 6.3.10.2.1.2

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

$- \frac{48 {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + \frac{8 {(2 x - 5)}^{3} (8 x^{2} - 5)}{{(8 x^{2} - 5)}^{3 + 1}}$

Step 6.3.10.2.2

Add $3$ and $1$ .

$- \frac{48 {(2 x - 5)}^{4} x}{{(8 x^{2} - 5)}^{4}} + \frac{8 {(2 x - 5)}^{3} (8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.3.11

Combine the numerators over the common denominator.

$\frac{- 48 {(2 x - 5)}^{4} x + 8 {(2 x - 5)}^{3} (8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.4

Reorder terms.

$\frac{- 48 x {(2 x - 5)}^{4} + 8 {(2 x - 5)}^{3} (8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Factor $8 {(2 x - 5)}^{3}$ out of $- 48 x {(2 x - 5)}^{4} + 8 {(2 x - 5)}^{3} (8 x^{2} - 5)$ .

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Step 6.5.1.1

Factor $8 {(2 x - 5)}^{3}$ out of $- 48 x {(2 x - 5)}^{4}$ .

$\frac{8 {(2 x - 5)}^{3} (- 6 x (2 x - 5)) + 8 {(2 x - 5)}^{3} (8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.1.2

Factor $8 {(2 x - 5)}^{3}$ out of $8 {(2 x - 5)}^{3} (- 6 x (2 x - 5)) + 8 {(2 x - 5)}^{3} (8 x^{2} - 5)$ .

$\frac{8 {(2 x - 5)}^{3} (- 6 x (2 x - 5) + 8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.2

Apply the distributive property.

$\frac{8 {(2 x - 5)}^{3} (- 6 x (2 x) - 6 x \cdot - 5 + 8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.3

Rewrite using the commutative property of multiplication.

$\frac{8 {(2 x - 5)}^{3} (- 6 \cdot 2 x \cdot x - 6 x \cdot - 5 + 8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.4

Multiply $- 5$ by $- 6$ .

$\frac{8 {(2 x - 5)}^{3} (- 6 \cdot 2 x \cdot x + 30 x + 8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.5

Simplify each term.

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Step 6.5.5.1

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

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Step 6.5.5.1.1

Move $x$ .

$\frac{8 {(2 x - 5)}^{3} (- 6 \cdot 2 (x \cdot x) + 30 x + 8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.5.1.2

Multiply $x$ by $x$ .

$\frac{8 {(2 x - 5)}^{3} (- 6 \cdot 2 x^{2} + 30 x + 8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.5.2

Multiply $- 6$ by $2$ .

$\frac{8 {(2 x - 5)}^{3} (- 12 x^{2} + 30 x + 8 x^{2} - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.5.6

Add $- 12 x^{2}$ and $8 x^{2}$ .

$\frac{8 {(2 x - 5)}^{3} (- 4 x^{2} + 30 x - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.6

Factor $- 1$ out of $- 4 x^{2}$ .

$\frac{8 {(2 x - 5)}^{3} (- (4 x^{2}) + 30 x - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.7

Factor $- 1$ out of $30 x$ .

$\frac{8 {(2 x - 5)}^{3} (- (4 x^{2}) - (- 30 x) - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.8

Factor $- 1$ out of $- (4 x^{2}) - (- 30 x)$ .

$\frac{8 {(2 x - 5)}^{3} (- (4 x^{2} - 30 x) - 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.9

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{8 {(2 x - 5)}^{3} (- (4 x^{2} - 30 x) - 1 (5))}{{(8 x^{2} - 5)}^{4}}$

Step 6.10

Factor $- 1$ out of $- (4 x^{2} - 30 x) - 1 (5)$ .

$\frac{8 {(2 x - 5)}^{3} (- (4 x^{2} - 30 x + 5))}{{(8 x^{2} - 5)}^{4}}$

Step 6.11

Rewrite $- (4 x^{2} - 30 x + 5)$ as $- 1 (4 x^{2} - 30 x + 5)$ .

$\frac{8 {(2 x - 5)}^{3} (- 1 (4 x^{2} - 30 x + 5))}{{(8 x^{2} - 5)}^{4}}$

Step 6.12

Move the negative in front of the fraction.

$- \frac{(8 {(2 x - 5)}^{3}) (4 x^{2} - 30 x + 5)}{{(8 x^{2} - 5)}^{4}}$

Step 6.13

Reorder factors in $- \frac{(8 {(2 x - 5)}^{3}) (4 x^{2} - 30 x + 5)}{{(8 x^{2} - 5)}^{4}}$ .

$- \frac{8 {(2 x - 5)}^{3} (4 x^{2} - 30 x + 5)}{{(8 x^{2} - 5)}^{4}}$