Solve (5x^1/2)(7x^1/5)=

$$( 5 x ^ { 1 / 2 } ) ( 7 x ^ { 1 / 5 } ) =$$

Evaluate

$$35x^{\frac{7}{10}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{5} to get \frac{7}{10}.

$$5x^{\frac{7}{10}}\times 7$$

Multiply 5 and 7 to get 35.

$$35x^{\frac{7}{10}}$$

Differentiate w.r.t. x

$$\frac{49}{2x^{\frac{3}{10}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and \frac{1}{5} to get \frac{7}{10}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(5x^{\frac{7}{10}}\times 7)$$

Multiply 5 and 7 to get 35.

$$\frac{\mathrm{d}}{\mathrm{d}x}(35x^{\frac{7}{10}})$$

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

$$\frac{7}{10}\times 35x^{\frac{7}{10}-1}$$

Multiply \frac{7}{10} times 35.

$$\frac{49}{2}x^{\frac{7}{10}-1}$$

Subtract 1 from \frac{7}{10}.

$$\frac{49}{2}x^{-\frac{3}{10}}$$