Solve [(x^3)^frac{1{3}}]^frac{1{5}}

$$[ ( x ^ { 3 } ) ^ { \frac { 1 } { 3 } } ] ^ { \frac { 1 } { 5 } }$$

Evaluate

$$\sqrt[5]{x}$$

Solution Steps

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

$$\left(x^{1}\right)^{\frac{1}{5}}$$

Calculate x to the power of 1 and get x.

$$x^{\frac{1}{5}}$$

Differentiate w.r.t. x

$$\frac{1}{5x^{\frac{4}{5}}}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(\left(x^{1}\right)^{\frac{1}{5}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}x}(x^{\frac{1}{5}})$$

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

$$\frac{1}{5}x^{\frac{1}{5}-1}$$

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

$$\frac{1}{5}x^{-\frac{4}{5}}$$