Calculus Examples

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

Step 1

By the Sum Rule, the derivative of $6 x^{4} - 5 x + x^{- \frac{1}{2}} (x^{\frac{3}{2}})$ with respect to $x$ is $\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$ .

$\frac{d}{d x} [6 x^{4}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$

Step 2

Evaluate $\frac{d}{d x} [6 x^{4}]$ .

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Step 2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{4}$ with respect to $x$ is $6 \frac{d}{d x} [x^{4}]$ .

$6 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$

Step 2.2

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

$6 (4 x^{3}) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$

Step 2.3

Multiply $4$ by $6$ .

$24 x^{3} + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$

Step 3

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

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Step 3.1

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

$24 x^{3} - 5 \frac{d}{d x} [x] + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$

Step 3.2

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

$24 x^{3} - 5 \cdot 1 + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$

Step 3.3

Multiply $- 5$ by $1$ .

$24 x^{3} - 5 + \frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$

Step 4

Evaluate $\frac{d}{d x} [x^{- \frac{1}{2}} (x^{\frac{3}{2}})]$ .

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Step 4.1

Multiply $x^{- \frac{1}{2}}$ by $x^{\frac{3}{2}}$ by adding the exponents.

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Step 4.1.1

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

$24 x^{3} - 5 + \frac{d}{d x} [x^{- \frac{1}{2} + \frac{3}{2}}]$

Step 4.1.2

Combine the numerators over the common denominator.

$24 x^{3} - 5 + \frac{d}{d x} [x^{\frac{- 1 + 3}{2}}]$

Step 4.1.3

Add $- 1$ and $3$ .

$24 x^{3} - 5 + \frac{d}{d x} [x^{\frac{2}{2}}]$

Step 4.1.4

Divide $2$ by $2$ .

$24 x^{3} - 5 + \frac{d}{d x} [x^{1}]$

Step 4.2

Simplify $x^{1}$ .

$24 x^{3} - 5 + \frac{d}{d x} [x]$

Step 4.3

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

$24 x^{3} - 5 + 1$

Step 5

Add $- 5$ and $1$ .

$24 x^{3} - 4$