Calculus Examples

$y = 5 x^{3} (4 x^{5} - 7 x^{3})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (5 x^{3} (4 x^{5} - 7 x^{3}))$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{3} (4 x^{5} - 7 x^{3})$ with respect to $x$ is $5 \frac{d}{d x} [x^{3} (4 x^{5} - 7 x^{3})]$ .

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

Step 3.2

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 $g (x) = 4 x^{5} - 7 x^{3}$ .

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

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $4 x^{5} - 7 x^{3}$ with respect to $x$ is $\frac{d}{d x} [4 x^{5}] + \frac{d}{d x} [- 7 x^{3}]$ .

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

Step 3.3.2

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

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $5$ by $4$ .

$5 (x^{3} (20 x^{4} + \frac{d}{d x} [- 7 x^{3}]) + (4 x^{5} - 7 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.3.5

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x^{3}$ with respect to $x$ is $- 7 \frac{d}{d x} [x^{3}]$ .

$5 (x^{3} (20 x^{4} - 7 \frac{d}{d x} [x^{3}]) + (4 x^{5} - 7 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.3.6

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

$5 (x^{3} (20 x^{4} - 7 (3 x^{2})) + (4 x^{5} - 7 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.3.7

Multiply $3$ by $- 7$ .

$5 (x^{3} (20 x^{4} - 21 x^{2}) + (4 x^{5} - 7 x^{3}) \frac{d}{d x} [x^{3}])$

Step 3.3.8

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

$5 (x^{3} (20 x^{4} - 21 x^{2}) + (4 x^{5} - 7 x^{3}) (3 x^{2}))$

Step 3.3.9

Move $3$ to the left of $4 x^{5} - 7 x^{3}$ .

$5 (x^{3} (20 x^{4} - 21 x^{2}) + 3 \cdot (4 x^{5} - 7 x^{3}) x^{2})$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$5 (x^{3} (20 x^{4}) + x^{3} (- 21 x^{2}) + 3 (4 x^{5} - 7 x^{3}) x^{2})$

Step 3.4.2

Apply the distributive property.

$5 (x^{3} (20 x^{4}) + x^{3} (- 21 x^{2}) + (3 (4 x^{5}) + 3 (- 7 x^{3})) x^{2})$

Step 3.4.3

Apply the distributive property.

$5 (x^{3} (20 x^{4}) + x^{3} (- 21 x^{2}) + 3 (4 x^{5}) x^{2} + 3 (- 7 x^{3}) x^{2})$

Step 3.4.4

Apply the distributive property.

$5 (x^{3} (20 x^{4})) + 5 (x^{3} (- 21 x^{2})) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5

Combine terms.

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Step 3.4.5.1

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

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Step 3.4.5.1.1

Move $x^{4}$ .

$5 (x^{4} x^{3} \cdot 20) + 5 (x^{3} (- 21 x^{2})) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.1.2

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

$5 (x^{4 + 3} \cdot 20) + 5 (x^{3} (- 21 x^{2})) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.1.3

Add $4$ and $3$ .

$5 (x^{7} \cdot 20) + 5 (x^{3} (- 21 x^{2})) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.2

Move $20$ to the left of $x^{7}$ .

$5 (20 \cdot x^{7}) + 5 (x^{3} (- 21 x^{2})) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.3

Multiply $20$ by $5$ .

$100 x^{7} + 5 (x^{3} (- 21 x^{2})) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.4

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.4.5.4.1

Move $x^{2}$ .

$100 x^{7} + 5 (x^{2} x^{3} \cdot - 21) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.4.2

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

$100 x^{7} + 5 (x^{2 + 3} \cdot - 21) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.4.3

Add $2$ and $3$ .

$100 x^{7} + 5 (x^{5} \cdot - 21) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.5

Move $- 21$ to the left of $x^{5}$ .

$100 x^{7} + 5 (- 21 \cdot x^{5}) + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.6

Multiply $- 21$ by $5$ .

$100 x^{7} - 105 x^{5} + 5 (3 (4 x^{5}) x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.7

Multiply $4$ by $3$ .

$100 x^{7} - 105 x^{5} + 5 (12 x^{5} x^{2}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.8

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

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Step 3.4.5.8.1

Move $x^{2}$ .

$100 x^{7} - 105 x^{5} + 5 (12 (x^{2} x^{5})) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.8.2

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

$100 x^{7} - 105 x^{5} + 5 (12 x^{2 + 5}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.8.3

Add $2$ and $5$ .

$100 x^{7} - 105 x^{5} + 5 (12 x^{7}) + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.9

Multiply $12$ by $5$ .

$100 x^{7} - 105 x^{5} + 60 x^{7} + 5 (3 (- 7 x^{3}) x^{2})$

Step 3.4.5.10

Multiply $- 7$ by $3$ .

$100 x^{7} - 105 x^{5} + 60 x^{7} + 5 (- 21 x^{3} x^{2})$

Step 3.4.5.11

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 3.4.5.11.1

Move $x^{2}$ .

$100 x^{7} - 105 x^{5} + 60 x^{7} + 5 (- 21 (x^{2} x^{3}))$

Step 3.4.5.11.2

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

$100 x^{7} - 105 x^{5} + 60 x^{7} + 5 (- 21 x^{2 + 3})$

Step 3.4.5.11.3

Add $2$ and $3$ .

$100 x^{7} - 105 x^{5} + 60 x^{7} + 5 (- 21 x^{5})$

Step 3.4.5.12

Multiply $- 21$ by $5$ .

$100 x^{7} - 105 x^{5} + 60 x^{7} - 105 x^{5}$

Step 3.4.5.13

Add $100 x^{7}$ and $60 x^{7}$ .

$160 x^{7} - 105 x^{5} - 105 x^{5}$

Step 3.4.5.14

Subtract $105 x^{5}$ from $- 105 x^{5}$ .

$160 x^{7} - 210 x^{5}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 160 x^{7} - 210 x^{5}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 160 x^{7} - 210 x^{5}$