Solve (-5x^frac{1{3}})(4x^-frac{3{4}})

$$( - 5 x ^ { \frac { 1 } { 3 } } ) ( 4 x ^ { - \frac { 3 } { 4 } } )$$

Evaluate

$$-\frac{20}{x^{\frac{5}{12}}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and -\frac{3}{4} to get -\frac{5}{12}.

$$-5x^{-\frac{5}{12}}\times 4$$

Multiply -5 and 4 to get -20.

$$-20x^{-\frac{5}{12}}$$

Differentiate w.r.t. x

$$\frac{25}{3x^{\frac{17}{12}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{3} and -\frac{3}{4} to get -\frac{5}{12}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-5x^{-\frac{5}{12}}\times 4)$$

Multiply -5 and 4 to get -20.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-20x^{-\frac{5}{12}})$$

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

$$-\frac{5}{12}\left(-20\right)x^{-\frac{5}{12}-1}$$

Multiply -\frac{5}{12} times -20.

$$\frac{25}{3}x^{-\frac{5}{12}-1}$$

Subtract 1 from -\frac{5}{12}.

$$\frac{25}{3}x^{-\frac{17}{12}}$$