Solve (5x^frac{1{9}})(3x^frac{2{5}})=

$$( 5 x ^ { \frac { 1 } { 9 } } ) ( 3 x ^ { \frac { 2 } { 5 } } ) =$$

Evaluate

$$15x^{\frac{23}{45}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{9} and \frac{2}{5} to get \frac{23}{45}.

$$5x^{\frac{23}{45}}\times 3$$

Multiply 5 and 3 to get 15.

$$15x^{\frac{23}{45}}$$

Differentiate w.r.t. x

$$\frac{23}{3x^{\frac{22}{45}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{9} and \frac{2}{5} to get \frac{23}{45}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{\frac{23}{45}}\times 3)$$

Multiply 5 and 3 to get 15.

$$\frac{\mathrm{d}}{\mathrm{d}x}(15x^{\frac{23}{45}})$$

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

$$\frac{23}{45}\times 15x^{\frac{23}{45}-1}$$

Multiply \frac{23}{45} times 15.

$$\frac{23}{3}x^{\frac{23}{45}-1}$$

Subtract 1 from \frac{23}{45}.

$$\frac{23}{3}x^{-\frac{22}{45}}$$