Find the Derivative - d/d@VAR f(x)=-9/7x^(7/6)

$f (x) = - \frac{9}{7} x^{\frac{7}{6}}$

Step 1

Since $- \frac{9}{7}$ is constant with respect to $x$ , the derivative of $- \frac{9}{7} x^{\frac{7}{6}}$ with respect to $x$ is $- \frac{9}{7} \frac{d}{d x} [x^{\frac{7}{6}}]$ .

$- \frac{9}{7} \frac{d}{d x} [x^{\frac{7}{6}}]$

Step 2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{7}{6}$ .

$- \frac{9}{7} (\frac{7}{6} x^{\frac{7}{6} - 1})$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- \frac{9}{7} (\frac{7}{6} x^{\frac{7}{6} - 1 \cdot \frac{6}{6}})$

Step 4

Combine $- 1$ and $\frac{6}{6}$ .

$- \frac{9}{7} (\frac{7}{6} x^{\frac{7}{6} + \frac{- 1 \cdot 6}{6}})$

Step 5

Combine the numerators over the common denominator.

$- \frac{9}{7} (\frac{7}{6} x^{\frac{7 - 1 \cdot 6}{6}})$

Step 6

Simplify the numerator.

Multiply $- 1$ by $6$ .

$- \frac{9}{7} (\frac{7}{6} x^{\frac{7 - 6}{6}})$

Subtract $6$ from $7$ .

$- \frac{9}{7} (\frac{7}{6} x^{\frac{1}{6}})$

Step 7

Combine $\frac{7}{6}$ and $x^{\frac{1}{6}}$ .

$- \frac{9}{7} \cdot \frac{7 x^{\frac{1}{6}}}{6}$

Step 8

Multiply $\frac{7 x^{\frac{1}{6}}}{6}$ by $\frac{9}{7}$ .

$- \frac{7 x^{\frac{1}{6}} \cdot 9}{6 \cdot 7}$

Step 9

Multiply.

Multiply $9$ by $7$ .

$- \frac{63 x^{\frac{1}{6}}}{6 \cdot 7}$

Multiply $6$ by $7$ .

$- \frac{63 x^{\frac{1}{6}}}{42}$

Step 10

Factor $21$ out of $63 x^{\frac{1}{6}}$ .

$- \frac{21 (3 x^{\frac{1}{6}})}{42}$

Step 11

Cancel the common factors.

Factor $21$ out of $42$ .

$- \frac{21 (3 x^{\frac{1}{6}})}{21 (2)}$

Cancel the common factor.

$- \frac{21 (3 x^{\frac{1}{6}})}{21 \cdot 2}$

Rewrite the expression.

$- \frac{3 x^{\frac{1}{6}}}{2}$

Find the Derivative - d/d@VAR f(x)=-9/7x^(7/6)

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