Algebra Examples

$f (x) = x^{6} - 5 x^{5} + 3 x^{2} + 2 - \sqrt{x^{3}}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $x^{6} - 5 x^{5} + 3 x^{2} + 2 - \sqrt{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 5 x^{5}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$ .

$\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- 5 x^{5}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$6 x^{5} + \frac{d}{d x} [- 5 x^{5}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 2

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

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Step 2.1

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

$6 x^{5} - 5 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 2.2

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

$6 x^{5} - 5 (5 x^{4}) + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 2.3

Multiply $5$ by $- 5$ .

$6 x^{5} - 25 x^{4} + \frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

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

$6 x^{5} - 25 x^{4} + 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 3.2

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

$6 x^{5} - 25 x^{4} + 3 (2 x) + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 3.3

Multiply $2$ by $3$ .

$6 x^{5} - 25 x^{4} + 6 x + \frac{d}{d x} [2] + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 4

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

$6 x^{5} - 25 x^{4} + 6 x + 0 + \frac{d}{d x} [- \sqrt{x^{3}}]$

Step 5

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

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Step 5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 + \frac{d}{d x} [- x^{\frac{3}{2}}]$

Step 5.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{3}{2}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{3}{2}}]$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 - \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 5.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 - (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 5.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 - (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 5.5

Combine $- 1$ and $\frac{2}{2}$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 - (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 5.6

Combine the numerators over the common denominator.

$6 x^{5} - 25 x^{4} + 6 x + 0 - (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 5.7

Simplify the numerator.

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Step 5.7.1

Multiply $- 1$ by $2$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 - (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 5.7.2

Subtract $2$ from $3$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 - (\frac{3}{2} x^{\frac{1}{2}})$

Step 5.8

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$6 x^{5} - 25 x^{4} + 6 x + 0 - \frac{3 x^{\frac{1}{2}}}{2}$

Step 6

Add $6 x^{5} - 25 x^{4} + 6 x$ and $0$ .

$6 x^{5} - 25 x^{4} + 6 x - \frac{3 x^{\frac{1}{2}}}{2}$