Calculus Examples

$y = {(2 x - 5)}^{- 1} {(x^{2} - 5 x)}^{6}$

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 $f (x) = {(2 x - 5)}^{- 1}$ and $g (x) = {(x^{2} - 5 x)}^{6}$ .

${(2 x - 5)}^{- 1} \frac{d}{d x} [{(x^{2} - 5 x)}^{6}] + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x^{2} - 5 x$ .

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Step 2.1

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

${(2 x - 5)}^{- 1} (\frac{d}{d u_{1}} [{u_{1}}^{6}] \frac{d}{d x} [x^{2} - 5 x]) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

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 $n = 6$ .

${(2 x - 5)}^{- 1} (6 {u_{1}}^{5} \frac{d}{d x} [x^{2} - 5 x]) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 2.3

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

${(2 x - 5)}^{- 1} (6 {(x^{2} - 5 x)}^{5} \frac{d}{d x} [x^{2} - 5 x]) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 3

Differentiate.

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Step 3.1

Move $6$ to the left of ${(2 x - 5)}^{- 1}$ .

$6 \cdot {(2 x - 5)}^{- 1} {(x^{2} - 5 x)}^{5} \frac{d}{d x} [x^{2} - 5 x] + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 3.2

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

$6 {(2 x - 5)}^{- 1} {(x^{2} - 5 x)}^{5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x]) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 3.3

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

$6 {(2 x - 5)}^{- 1} {(x^{2} - 5 x)}^{5} (2 x + \frac{d}{d x} [- 5 x]) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 3.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x$ with respect to $x$ is $- 5 \frac{d}{d x} [x]$ .

$6 {(2 x - 5)}^{- 1} {(x^{2} - 5 x)}^{5} (2 x - 5 \frac{d}{d x} [x]) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 3.5

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

$6 {(2 x - 5)}^{- 1} {(x^{2} - 5 x)}^{5} (2 x - 5 \cdot 1) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 3.6

Multiply $- 5$ by $1$ .

$6 {(2 x - 5)}^{- 1} {(x^{2} - 5 x)}^{5} (2 x - 5) + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 4

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

$6 ({(2 x - 5)}^{1} {(2 x - 5)}^{- 1}) {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 5

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

$6 {(2 x - 5)}^{1 - 1} {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 6

Simplify the expression.

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Step 6.1

Subtract $1$ from $1$ .

$6 {(2 x - 5)}^{0} {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 6.2

Anything raised to $0$ is $1$ .

$6 \cdot 1 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 6.3

Multiply $6$ by $1$ .

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} \frac{d}{d x} [{(2 x - 5)}^{- 1}]$

Step 7

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 2 x - 5$ .

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Step 7.1

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [2 x - 5])$

Step 7.2

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {u_{2}}^{- 2} \frac{d}{d x} [2 x - 5])$

Step 7.3

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {(2 x - 5)}^{- 2} \frac{d}{d x} [2 x - 5])$

Step 8

Differentiate.

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Step 8.1

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {(2 x - 5)}^{- 2} (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]))$

Step 8.2

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {(2 x - 5)}^{- 2} (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]))$

Step 8.3

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {(2 x - 5)}^{- 2} (2 \cdot 1 + \frac{d}{d x} [- 5]))$

Step 8.4

Multiply $2$ by $1$ .

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {(2 x - 5)}^{- 2} (2 + \frac{d}{d x} [- 5]))$

Step 8.5

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {(2 x - 5)}^{- 2} (2 + 0))$

Step 8.6

Simplify the expression.

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Step 8.6.1

Add $2$ and $0$ .

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- {(2 x - 5)}^{- 2} \cdot 2)$

Step 8.6.2

Multiply $2$ by $- 1$ .

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- 2 {(2 x - 5)}^{- 2})$

Step 9

Simplify.

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Step 9.1

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

$6 {(x^{2} - 5 x)}^{5} + {(x^{2} - 5 x)}^{6} (- 2 \frac{1}{{(2 x - 5)}^{2}})$

Step 9.2

Combine terms.

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Step 9.2.1

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

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

Step 9.2.2

Move the negative in front of the fraction.

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

Step 9.2.3

Combine ${(x^{2} - 5 x)}^{6}$ and $\frac{2}{{(2 x - 5)}^{2}}$ .

$6 {(x^{2} - 5 x)}^{5} - \frac{{(x^{2} - 5 x)}^{6} \cdot 2}{{(2 x - 5)}^{2}}$

Step 9.2.4

Move $2$ to the left of ${(x^{2} - 5 x)}^{6}$ .

$6 {(x^{2} - 5 x)}^{5} - \frac{2 \cdot {(x^{2} - 5 x)}^{6}}{{(2 x - 5)}^{2}}$

Step 9.2.5

To write $6 {(x^{2} - 5 x)}^{5}$ as a fraction with a common denominator, multiply by $\frac{{(2 x - 5)}^{2}}{{(2 x - 5)}^{2}}$ .

$\frac{6 {(x^{2} - 5 x)}^{5} {(2 x - 5)}^{2}}{{(2 x - 5)}^{2}} - \frac{2 {(x^{2} - 5 x)}^{6}}{{(2 x - 5)}^{2}}$

Step 9.2.6

Combine the numerators over the common denominator.

$\frac{6 {(x^{2} - 5 x)}^{5} {(2 x - 5)}^{2} - 2 {(x^{2} - 5 x)}^{6}}{{(2 x - 5)}^{2}}$

Step 9.3

Reorder terms.

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

Step 9.4

Simplify the numerator.

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Step 9.4.1

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

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Step 9.4.1.1

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

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

Step 9.4.1.2

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

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

Step 9.4.1.3

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

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

Step 9.4.2

Factor $x$ out of $x^{2} - 5 x$ .

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Step 9.4.2.1

Factor $x$ out of $x^{2}$ .

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

Step 9.4.2.2

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

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

Step 9.4.2.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

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

Step 9.4.3

Apply the product rule to $x (x - 5)$ .

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

Step 9.4.4

Simplify each term.

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Step 9.4.4.1

Apply the distributive property.

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

Step 9.4.4.2

Multiply $- 5$ by $- 1$ .

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

Step 9.4.4.3

Rewrite ${(2 x - 5)}^{2}$ as $(2 x - 5) (2 x - 5)$ .

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

Step 9.4.4.4

Expand $(2 x - 5) (2 x - 5)$ using the FOIL Method.

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Step 9.4.4.4.1

Apply the distributive property.

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

Step 9.4.4.4.2

Apply the distributive property.

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

Step 9.4.4.4.3

Apply the distributive property.

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

Step 9.4.4.5

Simplify and combine like terms.

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Step 9.4.4.5.1

Simplify each term.

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Step 9.4.4.5.1.1

Rewrite using the commutative property of multiplication.

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

Step 9.4.4.5.1.2

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

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Step 9.4.4.5.1.2.1

Move $x$ .

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

Step 9.4.4.5.1.2.2

Multiply $x$ by $x$ .

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

Step 9.4.4.5.1.3

Multiply $2$ by $2$ .

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

Step 9.4.4.5.1.4

Multiply $- 5$ by $2$ .

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 3 (4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5))}{{(2 x - 5)}^{2}}$

Step 9.4.4.5.1.5

Multiply $2$ by $- 5$ .

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 3 (4 x^{2} - 10 x - 10 x - 5 \cdot - 5))}{{(2 x - 5)}^{2}}$

Step 9.4.4.5.1.6

Multiply $- 5$ by $- 5$ .

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 3 (4 x^{2} - 10 x - 10 x + 25))}{{(2 x - 5)}^{2}}$

Step 9.4.4.5.2

Subtract $10 x$ from $- 10 x$ .

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 3 (4 x^{2} - 20 x + 25))}{{(2 x - 5)}^{2}}$

Step 9.4.4.6

Apply the distributive property.

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 3 (4 x^{2}) + 3 (- 20 x) + 3 \cdot 25)}{{(2 x - 5)}^{2}}$

Step 9.4.4.7

Simplify.

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Step 9.4.4.7.1

Multiply $4$ by $3$ .

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 12 x^{2} + 3 (- 20 x) + 3 \cdot 25)}{{(2 x - 5)}^{2}}$

Step 9.4.4.7.2

Multiply $- 20$ by $3$ .

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 12 x^{2} - 60 x + 3 \cdot 25)}{{(2 x - 5)}^{2}}$

Step 9.4.4.7.3

Multiply $3$ by $25$ .

$\frac{2 x^{5} {(x - 5)}^{5} (- x^{2} + 5 x + 12 x^{2} - 60 x + 75)}{{(2 x - 5)}^{2}}$

Step 9.4.5

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

$\frac{2 x^{5} {(x - 5)}^{5} (11 x^{2} + 5 x - 60 x + 75)}{{(2 x - 5)}^{2}}$

Step 9.4.6

Subtract $60 x$ from $5 x$ .

$\frac{2 x^{5} {(x - 5)}^{5} (11 x^{2} - 55 x + 75)}{{(2 x - 5)}^{2}}$