Solve frac{x^frac{3{20}}*x^frac{3{20}}}{x^1/10*x^1/40}=

$$\frac { x ^ { \frac { 3 } { 20 } } \cdot x ^ { \frac { 3 } { 20 } } } { x ^ { 1 / 10 } \cdot x ^ { 1 / 40 } } =$$

Evaluate

$$x^{\frac{7}{40}}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add \frac{3}{20} and \frac{3}{20} to get \frac{3}{10}.

$$\frac{x^{\frac{3}{10}}}{x^{\frac{1}{10}}x^{\frac{1}{40}}}$$

To multiply powers of the same base, add their exponents. Add \frac{1}{10} and \frac{1}{40} to get \frac{1}{8}.

$$\frac{x^{\frac{3}{10}}}{x^{\frac{1}{8}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{1}{8} from \frac{3}{10} to get \frac{7}{40}.

$$x^{\frac{7}{40}}$$

Differentiate w.r.t. x

$$\frac{7}{40x^{\frac{33}{40}}}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add \frac{3}{20} and \frac{3}{20} to get \frac{3}{10}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{3}{10}}}{x^{\frac{1}{10}}x^{\frac{1}{40}}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{10} and \frac{1}{40} to get \frac{1}{8}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{x^{\frac{3}{10}}}{x^{\frac{1}{8}}})$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{1}{8} from \frac{3}{10} to get \frac{7}{40}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{7}{40}})$$

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

$$\frac{7}{40}x^{\frac{7}{40}-1}$$

Subtract 1 from \frac{7}{40}.

$$\frac{7}{40}x^{-\frac{33}{40}}$$