Calculus Examples

$f (x) = x {(2 x - 5)}^{3}$

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

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

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

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Step 2.1

To apply the Chain Rule, set $u$ as $2 x - 5$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $2 x - 5$ .

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Multiply $2$ by $1$ .

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

Step 3.5

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

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

Step 3.6

Simplify the expression.

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Step 3.6.1

Add $2$ and $0$ .

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

Step 3.6.2

Multiply $2$ by $3$ .

$x (6 {(2 x - 5)}^{2}) + {(2 x - 5)}^{3} \frac{d}{d x} [x]$

Step 3.7

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

$x (6 {(2 x - 5)}^{2}) + {(2 x - 5)}^{3} \cdot 1$

Step 3.8

Multiply ${(2 x - 5)}^{3}$ by $1$ .

$x (6 {(2 x - 5)}^{2}) + {(2 x - 5)}^{3}$

Step 4

Simplify.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Combine terms.

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Step 4.2.1

Move $6$ to the left of $x$ .

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

Step 4.2.2

Add $6 x$ and $2 x$ .

${(2 x - 5)}^{2} (8 x - 5)$

Step 4.3

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

$(2 x - 5) (2 x - 5) (8 x - 5)$

Step 4.4

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

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Step 4.4.1

Apply the distributive property.

$(2 x (2 x - 5) - 5 (2 x - 5)) (8 x - 5)$

Step 4.4.2

Apply the distributive property.

$(2 x (2 x) + 2 x \cdot - 5 - 5 (2 x - 5)) (8 x - 5)$

Step 4.4.3

Apply the distributive property.

$(2 x (2 x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5) (8 x - 5)$

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Rewrite using the commutative property of multiplication.

$(2 \cdot 2 x \cdot x + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5) (8 x - 5)$

Step 4.5.1.2

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

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Step 4.5.1.2.1

Move $x$ .

$(2 \cdot 2 (x \cdot x) + 2 x \cdot - 5 - 5 (2 x) - 5 \cdot - 5) (8 x - 5)$

Step 4.5.1.2.2

Multiply $x$ by $x$ .

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

Step 4.5.1.3

Multiply $2$ by $2$ .

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

Step 4.5.1.4

Multiply $- 5$ by $2$ .

$(4 x^{2} - 10 x - 5 (2 x) - 5 \cdot - 5) (8 x - 5)$

Step 4.5.1.5

Multiply $2$ by $- 5$ .

$(4 x^{2} - 10 x - 10 x - 5 \cdot - 5) (8 x - 5)$

Step 4.5.1.6

Multiply $- 5$ by $- 5$ .

$(4 x^{2} - 10 x - 10 x + 25) (8 x - 5)$

Step 4.5.2

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

$(4 x^{2} - 20 x + 25) (8 x - 5)$

Step 4.6

Expand $(4 x^{2} - 20 x + 25) (8 x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$4 x^{2} (8 x) + 4 x^{2} \cdot - 5 - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7

Simplify each term.

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Step 4.7.1

Rewrite using the commutative property of multiplication.

$4 \cdot 8 x^{2} x + 4 x^{2} \cdot - 5 - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.7.2.1

Move $x$ .

$4 \cdot 8 (x \cdot x^{2}) + 4 x^{2} \cdot - 5 - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.2.2

Multiply $x$ by $x^{2}$ .

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Step 4.7.2.2.1

Raise $x$ to the power of $1$ .

$4 \cdot 8 (x^{1} x^{2}) + 4 x^{2} \cdot - 5 - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.2.2.2

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

$4 \cdot 8 x^{1 + 2} + 4 x^{2} \cdot - 5 - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.2.3

Add $1$ and $2$ .

$4 \cdot 8 x^{3} + 4 x^{2} \cdot - 5 - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.3

Multiply $4$ by $8$ .

$32 x^{3} + 4 x^{2} \cdot - 5 - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.4

Multiply $- 5$ by $4$ .

$32 x^{3} - 20 x^{2} - 20 x (8 x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.5

Rewrite using the commutative property of multiplication.

$32 x^{3} - 20 x^{2} - 20 \cdot 8 x \cdot x - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.6

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

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Step 4.7.6.1

Move $x$ .

$32 x^{3} - 20 x^{2} - 20 \cdot 8 (x \cdot x) - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.6.2

Multiply $x$ by $x$ .

$32 x^{3} - 20 x^{2} - 20 \cdot 8 x^{2} - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.7

Multiply $- 20$ by $8$ .

$32 x^{3} - 20 x^{2} - 160 x^{2} - 20 x \cdot - 5 + 25 (8 x) + 25 \cdot - 5$

Step 4.7.8

Multiply $- 5$ by $- 20$ .

$32 x^{3} - 20 x^{2} - 160 x^{2} + 100 x + 25 (8 x) + 25 \cdot - 5$

Step 4.7.9

Multiply $8$ by $25$ .

$32 x^{3} - 20 x^{2} - 160 x^{2} + 100 x + 200 x + 25 \cdot - 5$

Step 4.7.10

Multiply $25$ by $- 5$ .

$32 x^{3} - 20 x^{2} - 160 x^{2} + 100 x + 200 x - 125$

Step 4.8

Subtract $160 x^{2}$ from $- 20 x^{2}$ .

$32 x^{3} - 180 x^{2} + 100 x + 200 x - 125$

Step 4.9

Add $100 x$ and $200 x$ .

$32 x^{3} - 180 x^{2} + 300 x - 125$