Calculus Examples

$f (x) = 9 x^{\frac{7}{5}} - 5 x^{2} + 10000$

Step 1

By the Sum Rule, the derivative of $9 x^{\frac{7}{5}} - 5 x^{2} + 10000$ with respect to $x$ is $\frac{d}{d x} [9 x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$ .

$\frac{d}{d x} [9 x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2

Evaluate $\frac{d}{d x} [9 x^{\frac{7}{5}}]$ .

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Step 2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{\frac{7}{5}}$ with respect to $x$ is $9 \frac{d}{d x} [x^{\frac{7}{5}}]$ .

$9 \frac{d}{d x} [x^{\frac{7}{5}}] + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} - 1}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} - 1 \cdot \frac{5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.4

Combine $- 1$ and $\frac{5}{5}$ .

$9 (\frac{7}{5} x^{\frac{7}{5} + \frac{- 1 \cdot 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.5

Combine the numerators over the common denominator.

$9 (\frac{7}{5} x^{\frac{7 - 1 \cdot 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $5$ .

$9 (\frac{7}{5} x^{\frac{7 - 5}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.6.2

Subtract $5$ from $7$ .

$9 (\frac{7}{5} x^{\frac{2}{5}}) + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.7

Combine $\frac{7}{5}$ and $x^{\frac{2}{5}}$ .

$9 \frac{7 x^{\frac{2}{5}}}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.8

Combine $9$ and $\frac{7 x^{\frac{2}{5}}}{5}$ .

$\frac{9 (7 x^{\frac{2}{5}})}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 2.9

Multiply $7$ by $9$ .

$\frac{63 x^{\frac{2}{5}}}{5} + \frac{d}{d x} [- 5 x^{2}] + \frac{d}{d x} [10000]$

Step 3

Evaluate $\frac{d}{d x} [- 5 x^{2}]$ .

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Step 3.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 5 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [10000]$

Step 3.2

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

$\frac{63 x^{\frac{2}{5}}}{5} - 5 (2 x) + \frac{d}{d x} [10000]$

Step 3.3

Multiply $2$ by $- 5$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x + \frac{d}{d x} [10000]$

Step 4

Since $10000$ is constant with respect to $x$ , the derivative of $10000$ with respect to $x$ is $0$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x + 0$

Step 5

Simplify.

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Step 5.1

Add $\frac{63 x^{\frac{2}{5}}}{5} - 10 x$ and $0$ .

$\frac{63 x^{\frac{2}{5}}}{5} - 10 x$

Step 5.2

Reorder terms.

$- 10 x + \frac{63 x^{\frac{2}{5}}}{5}$