Solve -(5x^frac{1{2}})(4/3x^frac{1{3}})

$$- ( 5 x ^ { \frac { 1 } { 2 } } ) ( \frac { 4 } { 3 } x ^ { \frac { 1 } { 3 } } )$$

Evaluate

$$-\frac{20x^{\frac{5}{6}}}{3}$$

Solution Steps

Multiply -1 and 5 to get -5.

$$-5x^{\frac{1}{2}}\times \frac{4}{3}x^{\frac{1}{3}}$$

Multiply -5 and \frac{4}{3} to get -\frac{20}{3}.

$$-\frac{20}{3}x^{\frac{1}{2}}x^{\frac{1}{3}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{3} to get \frac{5}{6}.

$$-\frac{20}{3}x^{\frac{5}{6}}$$

Differentiate w.r.t. x

$$-\frac{50}{9\sqrt[6]{x}}$$

Steps Using Definition of a Derivative

Multiply -1 and 5 to get -5.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-5x^{\frac{1}{2}}\times \frac{4}{3}x^{\frac{1}{3}})$$

Multiply -5 and \frac{4}{3} to get -\frac{20}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\frac{20}{3}x^{\frac{1}{2}}x^{\frac{1}{3}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{3} to get \frac{5}{6}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(-\frac{20}{3}x^{\frac{5}{6}})$$

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

$$\frac{5}{6}\left(-\frac{20}{3}\right)x^{\frac{5}{6}-1}$$

Multiply \frac{5}{6} times -\frac{20}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-\frac{50}{9}x^{\frac{5}{6}-1}$$

Subtract 1 from \frac{5}{6}.

$$-\frac{50}{9}x^{-\frac{1}{6}}$$