Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $5 x - 4 x^{2}$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [- 4 x^{2}]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Multiply $5$ by $1$ .

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

Step 2.5

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

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

Step 2.6

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

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

Step 2.7

Multiply $2$ by $- 4$ .

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

Step 2.8

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

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

Step 2.9

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 3.4

Combine terms.

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Step 3.4.1

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

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

Step 3.4.2

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

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Step 3.4.2.1

Move $x$ .

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

Step 3.4.2.2

Multiply $x$ by $x^{4}$ .

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Step 3.4.2.2.1

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

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

Step 3.4.2.2.2

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

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

Step 3.4.2.3

Add $1$ and $4$ .

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

Step 3.4.3

Move $- 8$ to the left of $x^{5}$ .

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

Step 3.4.4

Multiply $5$ by $4$ .

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

Step 3.4.5

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

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Step 3.4.5.1

Move $x^{3}$ .

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

Step 3.4.5.2

Multiply $x^{3}$ by $x$ .

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Step 3.4.5.2.1

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

$5 x^{4} - 8 x^{5} + 20 (x^{3} x^{1}) + 4 (- 4 x^{2}) x^{3}$

Step 3.4.5.2.2

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

$5 x^{4} - 8 x^{5} + 20 x^{3 + 1} + 4 (- 4 x^{2}) x^{3}$

Step 3.4.5.3

Add $3$ and $1$ .

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

Step 3.4.6

Multiply $- 4$ by $4$ .

$5 x^{4} - 8 x^{5} + 20 x^{4} - 16 x^{2} x^{3}$

Step 3.4.7

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

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Step 3.4.7.1

Move $x^{3}$ .

$5 x^{4} - 8 x^{5} + 20 x^{4} - 16 (x^{3} x^{2})$

Step 3.4.7.2

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

$5 x^{4} - 8 x^{5} + 20 x^{4} - 16 x^{3 + 2}$

Step 3.4.7.3

Add $3$ and $2$ .

$5 x^{4} - 8 x^{5} + 20 x^{4} - 16 x^{5}$

Step 3.4.8

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

$- 8 x^{5} + 25 x^{4} - 16 x^{5}$

Step 3.4.9

Subtract $16 x^{5}$ from $- 8 x^{5}$ .

$- 24 x^{5} + 25 x^{4}$