Solve x^frac{4{7}}*(x^frac{2{3}})^frac{1{5}}

$$x ^ { \frac { 4 } { 7 } } \cdot ( x ^ { \frac { 2 } { 3 } } ) ^ { \frac { 1 } { 5 } }$$

Evaluate

$$x^{\frac{74}{105}}$$

Solution Steps

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and \frac{1}{5} to get \frac{2}{15}.

$$x^{\frac{4}{7}}x^{\frac{2}{15}}$$

To multiply powers of the same base, add their exponents. Add \frac{4}{7} and \frac{2}{15} to get \frac{74}{105}.

$$x^{\frac{74}{105}}$$

Differentiate w.r.t. x

$$\frac{74}{105x^{\frac{31}{105}}}$$

Steps Using Definition of a Derivative

To raise a power to another power, multiply the exponents. Multiply \frac{2}{3} and \frac{1}{5} to get \frac{2}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{4}{7}}x^{\frac{2}{15}})$$

To multiply powers of the same base, add their exponents. Add \frac{4}{7} and \frac{2}{15} to get \frac{74}{105}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{74}{105}})$$

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

$$\frac{74}{105}x^{\frac{74}{105}-1}$$

Subtract 1 from \frac{74}{105}.

$$\frac{74}{105}x^{-\frac{31}{105}}$$