Solve 12x^-frac{1{5}}div3x^-frac{6{5}}

$$12 x ^ { - \frac { 1 } { 5 } } \div 3 x ^ { - \frac { 6 } { 5 } }$$

Evaluate

$$\frac{4}{x^{\frac{7}{5}}}$$

Solution Steps

Divide 12x^{-\frac{1}{5}} by 3 to get 4x^{-\frac{1}{5}}.

$$4x^{-\frac{1}{5}}x^{-\frac{6}{5}}$$

To multiply powers of the same base, add their exponents. Add -\frac{1}{5} and -\frac{6}{5} to get -\frac{7}{5}.

$$4x^{-\frac{7}{5}}$$

Differentiate w.r.t. x

$$-\frac{28}{5x^{\frac{12}{5}}}$$

Steps Using Definition of a Derivative

Divide 12x^{-\frac{1}{5}} by 3 to get 4x^{-\frac{1}{5}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{-\frac{1}{5}}x^{-\frac{6}{5}})$$

To multiply powers of the same base, add their exponents. Add -\frac{1}{5} and -\frac{6}{5} to get -\frac{7}{5}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(4x^{-\frac{7}{5}})$$

The derivative of ax^{n} is nax^{n-1}.

$$-\frac{7}{5}\times 4x^{-\frac{7}{5}-1}$$

Multiply -\frac{7}{5} times 4.

$$-\frac{28}{5}x^{-\frac{7}{5}-1}$$

Subtract 1 from -\frac{7}{5}.

$$-\frac{28}{5}x^{-\frac{12}{5}}$$