Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $x^{3}$ and $\frac{5}{2}$ .

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

Step 1.2

Combine fractions.

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Step 1.2.1

Combine $\frac{x^{3} \cdot 5}{2}$ and ${(4 x^{5} - 5)}^{3}$ .

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

Step 1.2.2

Move $5$ to the left of $x^{3}$ .

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

Step 1.3

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5 x^{3} {(4 x^{5} - 5)}^{3}}{2}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [x^{3} {(4 x^{5} - 5)}^{3}]$ .

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

Step 2

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

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 4 x^{5} - 5$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $4 x^{5} - 5$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $4 x^{5} - 5$ .

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

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $4 x^{5} - 5$ with respect to $x$ is $\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [- 5]$ .

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

Step 4.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{5}$ with respect to $x$ is $4 \frac{d}{d x} [x^{5}]$ .

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

Step 4.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

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

Step 4.4

Multiply $5$ by $4$ .

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

Step 4.5

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

$\frac{5}{2} (x^{3} (3 {(4 x^{5} - 5)}^{2} (20 x^{4} + 0)) + {(4 x^{5} - 5)}^{3} \frac{d}{d x} [x^{3}])$

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $20 x^{4}$ and $0$ .

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

Step 4.6.2

Multiply $20$ by $3$ .

$\frac{5}{2} (x^{3} (60 {(4 x^{5} - 5)}^{2} x^{4}) + {(4 x^{5} - 5)}^{3} \frac{d}{d x} [x^{3}])$

Step 5

Multiply $x^{3}$ by $x^{4}$ by adding the exponents.

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Step 5.1

Move $x^{4}$ .

$\frac{5}{2} (x^{4} x^{3} (60 {(4 x^{5} - 5)}^{2}) + {(4 x^{5} - 5)}^{3} \frac{d}{d x} [x^{3}])$

Step 5.2

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

$\frac{5}{2} (x^{4 + 3} (60 {(4 x^{5} - 5)}^{2}) + {(4 x^{5} - 5)}^{3} \frac{d}{d x} [x^{3}])$

Step 5.3

Add $4$ and $3$ .

$\frac{5}{2} (x^{7} (60 {(4 x^{5} - 5)}^{2}) + {(4 x^{5} - 5)}^{3} \frac{d}{d x} [x^{3}])$

Step 6

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

$\frac{5}{2} (x^{7} (60 {(4 x^{5} - 5)}^{2}) + {(4 x^{5} - 5)}^{3} (3 x^{2}))$

Step 7

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

$\frac{5}{2} (x^{7} (60 {(4 x^{5} - 5)}^{2}) + 3 \cdot {(4 x^{5} - 5)}^{3} x^{2})$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$\frac{5}{2} (x^{7} (60 {(4 x^{5} - 5)}^{2})) + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2

Combine terms.

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Step 8.2.1

Combine $x^{7}$ and $\frac{5}{2}$ .

$\frac{x^{7} \cdot 5}{2} (60 {(4 x^{5} - 5)}^{2}) + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.2

Combine $60$ and $\frac{x^{7} \cdot 5}{2}$ .

$\frac{60 (x^{7} \cdot 5)}{2} {(4 x^{5} - 5)}^{2} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.3

Multiply $5$ by $60$ .

$\frac{300 x^{7}}{2} {(4 x^{5} - 5)}^{2} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.4

Combine $\frac{300 x^{7}}{2}$ and ${(4 x^{5} - 5)}^{2}$ .

$\frac{300 x^{7} {(4 x^{5} - 5)}^{2}}{2} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.5

Cancel the common factor of $300$ and $2$ .

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Step 8.2.5.1

Factor $2$ out of $300 x^{7} {(4 x^{5} - 5)}^{2}$ .

$\frac{2 (150 x^{7} {(4 x^{5} - 5)}^{2})}{2} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.5.2

Cancel the common factors.

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Step 8.2.5.2.1

Factor $2$ out of $2$ .

$\frac{2 (150 x^{7} {(4 x^{5} - 5)}^{2})}{2 (1)} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.5.2.2

Cancel the common factor.

$\frac{2 (150 x^{7} {(4 x^{5} - 5)}^{2})}{2 \cdot 1} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.5.2.3

Rewrite the expression.

$\frac{150 x^{7} {(4 x^{5} - 5)}^{2}}{1} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.5.2.4

Divide $150 x^{7} {(4 x^{5} - 5)}^{2}$ by $1$ .

$150 x^{7} {(4 x^{5} - 5)}^{2} + \frac{5}{2} (3 {(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.6

Combine $3$ and $\frac{5}{2}$ .

$150 x^{7} {(4 x^{5} - 5)}^{2} + \frac{3 \cdot 5}{2} ({(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.7

Multiply $3$ by $5$ .

$150 x^{7} {(4 x^{5} - 5)}^{2} + \frac{15}{2} ({(4 x^{5} - 5)}^{3} x^{2})$

Step 8.2.8

Combine ${(4 x^{5} - 5)}^{3}$ and $\frac{15}{2}$ .

$150 x^{7} {(4 x^{5} - 5)}^{2} + \frac{{(4 x^{5} - 5)}^{3} \cdot 15}{2} x^{2}$

Step 8.2.9

Combine $\frac{{(4 x^{5} - 5)}^{3} \cdot 15}{2}$ and $x^{2}$ .

$150 x^{7} {(4 x^{5} - 5)}^{2} + \frac{{(4 x^{5} - 5)}^{3} \cdot 15 x^{2}}{2}$

Step 8.2.10

Move $15$ to the left of ${(4 x^{5} - 5)}^{3}$ .

$150 x^{7} {(4 x^{5} - 5)}^{2} + \frac{15 \cdot {(4 x^{5} - 5)}^{3} x^{2}}{2}$

Step 8.2.11

To write $150 x^{7} {(4 x^{5} - 5)}^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$150 x^{7} {(4 x^{5} - 5)}^{2} \cdot \frac{2}{2} + \frac{15 {(4 x^{5} - 5)}^{3} x^{2}}{2}$

Step 8.2.12

Combine $150 x^{7} {(4 x^{5} - 5)}^{2}$ and $\frac{2}{2}$ .

$\frac{150 x^{7} {(4 x^{5} - 5)}^{2} \cdot 2}{2} + \frac{15 {(4 x^{5} - 5)}^{3} x^{2}}{2}$

Step 8.2.13

Combine the numerators over the common denominator.

$\frac{150 x^{7} {(4 x^{5} - 5)}^{2} \cdot 2 + 15 {(4 x^{5} - 5)}^{3} x^{2}}{2}$

Step 8.2.14

Multiply $2$ by $150$ .

$\frac{300 x^{7} {(4 x^{5} - 5)}^{2} + 15 {(4 x^{5} - 5)}^{3} x^{2}}{2}$

Step 8.3

Reorder terms.

$\frac{300 x^{7} {(4 x^{5} - 5)}^{2} + 15 x^{2} {(4 x^{5} - 5)}^{3}}{2}$

Step 8.4

Simplify the numerator.

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Step 8.4.1

Factor $15 x^{2} {(4 x^{5} - 5)}^{2}$ out of $300 x^{7} {(4 x^{5} - 5)}^{2} + 15 x^{2} {(4 x^{5} - 5)}^{3}$ .

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Step 8.4.1.1

Factor $15 x^{2} {(4 x^{5} - 5)}^{2}$ out of $300 x^{7} {(4 x^{5} - 5)}^{2}$ .

$\frac{15 x^{2} {(4 x^{5} - 5)}^{2} (20 x^{5}) + 15 x^{2} {(4 x^{5} - 5)}^{3}}{2}$

Step 8.4.1.2

Factor $15 x^{2} {(4 x^{5} - 5)}^{2}$ out of $15 x^{2} {(4 x^{5} - 5)}^{3}$ .

$\frac{15 x^{2} {(4 x^{5} - 5)}^{2} (20 x^{5}) + 15 x^{2} {(4 x^{5} - 5)}^{2} (4 x^{5} - 5)}{2}$

Step 8.4.1.3

Factor $15 x^{2} {(4 x^{5} - 5)}^{2}$ out of $15 x^{2} {(4 x^{5} - 5)}^{2} (20 x^{5}) + 15 x^{2} {(4 x^{5} - 5)}^{2} (4 x^{5} - 5)$ .

$\frac{15 x^{2} {(4 x^{5} - 5)}^{2} (20 x^{5} + 4 x^{5} - 5)}{2}$

Step 8.4.2

Add $20 x^{5}$ and $4 x^{5}$ .

$\frac{15 x^{2} {(4 x^{5} - 5)}^{2} (24 x^{5} - 5)}{2}$