Solve (64s^frac{3{7}})^frac{1{6}}

$$( 64 s ^ { \frac { 3 } { 7 } } ) ^ { \frac { 1 } { 6 } }$$

Evaluate

$$2\sqrt[14]{s}$$

Solution Steps

Expand \left(64s^{\frac{3}{7}}\right)^{\frac{1}{6}}.

$$64^{\frac{1}{6}}\left(s^{\frac{3}{7}}\right)^{\frac{1}{6}}$$

To raise a power to another power, multiply the exponents. Multiply \frac{3}{7} and \frac{1}{6} to get \frac{1}{14}.

$$64^{\frac{1}{6}}s^{\frac{1}{14}}$$

Calculate 64 to the power of \frac{1}{6} and get 2.

$$2s^{\frac{1}{14}}$$

Differentiate w.r.t. s

$$\frac{1}{7s^{\frac{13}{14}}}$$

Steps Using Chain Rule

If F is the composition of two differentiable functions f\left(u\right) and u=g\left(x\right), that is, if F\left(x\right)=f\left(g\left(x\right)\right), then the derivative of F is the derivative of f with respect to u times the derivative of g with respect to x, that is, \frac{\mathrm{d}}{\mathrm{d}x}(F)\left(x\right)=\frac{\mathrm{d}}{\mathrm{d}x}(f)\left(g\left(x\right)\right)\frac{\mathrm{d}}{\mathrm{d}x}(g)\left(x\right).

$$\frac{1}{6}\times \left(64s^{\frac{3}{7}}\right)^{\frac{1}{6}-1}\frac{\mathrm{d}}{\mathrm{d}s}(64s^{\frac{3}{7}})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$\frac{1}{6}\times \left(64s^{\frac{3}{7}}\right)^{-\frac{5}{6}}\times \frac{3}{7}\times 64s^{\frac{3}{7}-1}$$

Simplify.

$$\frac{32}{7}s^{-\frac{4}{7}}\times \left(64s^{\frac{3}{7}}\right)^{-\frac{5}{6}}$$