Calculus Examples

$(5 x^{4} - x + 2) (- x^{5} + 9)$

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 $f (x) = 5 x^{4} - x + 2$ and $g (x) = - x^{5} + 9$ .

$(5 x^{4} - x + 2) \frac{d}{d x} [- x^{5} + 9] + (- x^{5} + 9) \frac{d}{d x} [5 x^{4} - x + 2]$

Step 2

Differentiate.

Tap for more steps...

Step 2.1

By the Sum Rule, the derivative of $- x^{5} + 9$ with respect to $x$ is $\frac{d}{d x} [- x^{5}] + \frac{d}{d x} [9]$ .

$(5 x^{4} - x + 2) (\frac{d}{d x} [- x^{5}] + \frac{d}{d x} [9]) + (- x^{5} + 9) \frac{d}{d x} [5 x^{4} - x + 2]$

Step 2.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{5}$ with respect to $x$ is $- \frac{d}{d x} [x^{5}]$ .

$(5 x^{4} - x + 2) (- \frac{d}{d x} [x^{5}] + \frac{d}{d x} [9]) + (- x^{5} + 9) \frac{d}{d x} [5 x^{4} - x + 2]$

Step 2.3

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

$(5 x^{4} - x + 2) (- (5 x^{4}) + \frac{d}{d x} [9]) + (- x^{5} + 9) \frac{d}{d x} [5 x^{4} - x + 2]$

Step 2.4

Multiply $5$ by $- 1$ .

$(5 x^{4} - x + 2) (- 5 x^{4} + \frac{d}{d x} [9]) + (- x^{5} + 9) \frac{d}{d x} [5 x^{4} - x + 2]$

Step 2.5

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$(5 x^{4} - x + 2) (- 5 x^{4} + 0) + (- x^{5} + 9) \frac{d}{d x} [5 x^{4} - x + 2]$

Step 2.6

Add $- 5 x^{4}$ and $0$ .

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) \frac{d}{d x} [5 x^{4} - x + 2]$

Step 2.7

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

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [- x] + \frac{d}{d x} [2])$

Step 2.8

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

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- x] + \frac{d}{d x} [2])$

Step 2.9

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

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (5 (4 x^{3}) + \frac{d}{d x} [- x] + \frac{d}{d x} [2])$

Step 2.10

Multiply $4$ by $5$ .

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3} + \frac{d}{d x} [- x] + \frac{d}{d x} [2])$

Step 2.11

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3} - \frac{d}{d x} [x] + \frac{d}{d x} [2])$

Step 2.12

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

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3} - 1 \cdot 1 + \frac{d}{d x} [2])$

Step 2.13

Multiply $- 1$ by $1$ .

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3} - 1 + \frac{d}{d x} [2])$

Step 2.14

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3} - 1 + 0)$

Step 2.15

Add $20 x^{3} - 1$ and $0$ .

$(5 x^{4} - x + 2) (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3} - 1)$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$5 x^{4} (- 5 x^{4}) - x (- 5 x^{4}) + 2 (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3} - 1)$

Step 3.2

Apply the distributive property.

$5 x^{4} (- 5 x^{4}) - x (- 5 x^{4}) + 2 (- 5 x^{4}) + (- x^{5} + 9) (20 x^{3}) + (- x^{5} + 9) \cdot - 1$

Step 3.3

Apply the distributive property.

$5 x^{4} (- 5 x^{4}) - x (- 5 x^{4}) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) + (- x^{5} + 9) \cdot - 1$

Step 3.4

Apply the distributive property.

$5 x^{4} (- 5 x^{4}) - x (- 5 x^{4}) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5

Combine terms.

Tap for more steps...

Step 3.5.1

Multiply $- 5$ by $5$ .

$- 25 x^{4} x^{4} - x (- 5 x^{4}) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.2

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

Tap for more steps...

Step 3.5.2.1

Move $x^{4}$ .

$- 25 (x^{4} x^{4}) - x (- 5 x^{4}) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.2.2

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

$- 25 x^{4 + 4} - x (- 5 x^{4}) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.2.3

Add $4$ and $4$ .

$- 25 x^{8} - x (- 5 x^{4}) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.3

Multiply $- 5$ by $- 1$ .

$- 25 x^{8} + 5 x \cdot x^{4} + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.4

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

Tap for more steps...

Step 3.5.4.1

Move $x^{4}$ .

$- 25 x^{8} + 5 (x^{4} x) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.4.2

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

Tap for more steps...

Step 3.5.4.2.1

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

$- 25 x^{8} + 5 (x^{4} x^{1}) + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.4.2.2

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

$- 25 x^{8} + 5 x^{4 + 1} + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.4.3

Add $4$ and $1$ .

$- 25 x^{8} + 5 x^{5} + 2 (- 5 x^{4}) - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.5

Multiply $- 5$ by $2$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - x^{5} (20 x^{3}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.6

Multiply $20$ by $- 1$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 x^{5} x^{3} + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.7

Multiply $x^{5}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 3.5.7.1

Move $x^{3}$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 (x^{3} x^{5}) + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.7.2

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

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 x^{3 + 5} + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.7.3

Add $3$ and $5$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 x^{8} + 9 (20 x^{3}) - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.8

Multiply $20$ by $9$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 x^{8} + 180 x^{3} - x^{5} \cdot - 1 + 9 \cdot - 1$

Step 3.5.9

Multiply $- 1$ by $- 1$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 x^{8} + 180 x^{3} + 1 x^{5} + 9 \cdot - 1$

Step 3.5.10

Multiply $x^{5}$ by $1$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 x^{8} + 180 x^{3} + x^{5} + 9 \cdot - 1$

Step 3.5.11

Multiply $9$ by $- 1$ .

$- 25 x^{8} + 5 x^{5} - 10 x^{4} - 20 x^{8} + 180 x^{3} + x^{5} - 9$

Step 3.5.12

Subtract $20 x^{8}$ from $- 25 x^{8}$ .

$- 45 x^{8} + 5 x^{5} - 10 x^{4} + 180 x^{3} + x^{5} - 9$

Step 3.5.13

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

$- 45 x^{8} + 6 x^{5} - 10 x^{4} + 180 x^{3} - 9$