Calculus Examples

$y = - x^{5} + 10 x^{3} - 4 x^{2} - 25 x + 20$

Step 1

By the Sum Rule, the derivative of $- x^{5} + 10 x^{3} - 4 x^{2} - 25 x + 20$ with respect to $x$ is $\frac{d}{d x} [- x^{5}] + \frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$ .

$\frac{d}{d x} [- x^{5}] + \frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 2

Evaluate $\frac{d}{d x} [- x^{5}]$ .

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Step 2.1

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

$- \frac{d}{d x} [x^{5}] + \frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 2.2

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

$- (5 x^{4}) + \frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 2.3

Multiply $5$ by $- 1$ .

$- 5 x^{4} + \frac{d}{d x} [10 x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 3

Evaluate $\frac{d}{d x} [10 x^{3}]$ .

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Step 3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{3}$ with respect to $x$ is $10 \frac{d}{d x} [x^{3}]$ .

$- 5 x^{4} + 10 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 3.2

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

$- 5 x^{4} + 10 (3 x^{2}) + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 3.3

Multiply $3$ by $10$ .

$- 5 x^{4} + 30 x^{2} + \frac{d}{d x} [- 4 x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 4

Evaluate $\frac{d}{d x} [- 4 x^{2}]$ .

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Step 4.1

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

$- 5 x^{4} + 30 x^{2} - 4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 4.2

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

$- 5 x^{4} + 30 x^{2} - 4 (2 x) + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 4.3

Multiply $2$ by $- 4$ .

$- 5 x^{4} + 30 x^{2} - 8 x + \frac{d}{d x} [- 25 x] + \frac{d}{d x} [20]$

Step 5

Evaluate $\frac{d}{d x} [- 25 x]$ .

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Step 5.1

Since $- 25$ is constant with respect to $x$ , the derivative of $- 25 x$ with respect to $x$ is $- 25 \frac{d}{d x} [x]$ .

$- 5 x^{4} + 30 x^{2} - 8 x - 25 \frac{d}{d x} [x] + \frac{d}{d x} [20]$

Step 5.2

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

$- 5 x^{4} + 30 x^{2} - 8 x - 25 \cdot 1 + \frac{d}{d x} [20]$

Step 5.3

Multiply $- 25$ by $1$ .

$- 5 x^{4} + 30 x^{2} - 8 x - 25 + \frac{d}{d x} [20]$

Step 6

Differentiate using the Constant Rule.

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Step 6.1

Since $20$ is constant with respect to $x$ , the derivative of $20$ with respect to $x$ is $0$ .

$- 5 x^{4} + 30 x^{2} - 8 x - 25 + 0$

Step 6.2

Add $- 5 x^{4} + 30 x^{2} - 8 x - 25$ and $0$ .

$- 5 x^{4} + 30 x^{2} - 8 x - 25$