Calculus Examples

$11 x^{5} - 3 x^{2} + 9 x - π$

Step 1

By the Sum Rule, the derivative of $11 x^{5} - 3 x^{2} + 9 x - π$ with respect to $x$ is $\frac{d}{d x} [11 x^{5}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$ .

$\frac{d}{d x} [11 x^{5}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$

Step 2

Evaluate $\frac{d}{d x} [11 x^{5}]$ .

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Step 2.1

Since $11$ is constant with respect to $x$ , the derivative of $11 x^{5}$ with respect to $x$ is $11 \frac{d}{d x} [x^{5}]$ .

$11 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$

Step 2.2

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

$11 (5 x^{4}) + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$

Step 2.3

Multiply $5$ by $11$ .

$55 x^{4} + \frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$

Step 3

Evaluate $\frac{d}{d x} [- 3 x^{2}]$ .

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Step 3.1

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

$55 x^{4} - 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$

Step 3.2

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

$55 x^{4} - 3 (2 x) + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$

Step 3.3

Multiply $2$ by $- 3$ .

$55 x^{4} - 6 x + \frac{d}{d x} [9 x] + \frac{d}{d x} [- π]$

Step 4

Evaluate $\frac{d}{d x} [9 x]$ .

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Step 4.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x$ with respect to $x$ is $9 \frac{d}{d x} [x]$ .

$55 x^{4} - 6 x + 9 \frac{d}{d x} [x] + \frac{d}{d x} [- π]$

Step 4.2

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

$55 x^{4} - 6 x + 9 \cdot 1 + \frac{d}{d x} [- π]$

Step 4.3

Multiply $9$ by $1$ .

$55 x^{4} - 6 x + 9 + \frac{d}{d x} [- π]$

Step 5

Differentiate using the Constant Rule.

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Step 5.1

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

$55 x^{4} - 6 x + 9 + 0$

Step 5.2

Add $55 x^{4} - 6 x + 9$ and $0$ .

$55 x^{4} - 6 x + 9$