Solve (w^frac{12{7}})^frac{5{9}}

$$( w ^ { \frac { 12 } { 7 } } ) ^ { \frac { 5 } { 9 } }$$

Differentiate w.r.t. w

$$\frac{20}{21\sqrt[21]{w}}$$

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).

$$\frac{5}{9}\left(w^{\frac{12}{7}}\right)^{\frac{5}{9}-1}\frac{\mathrm{d}}{\mathrm{d}w}(w^{\frac{12}{7}})$$

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}.

$$\frac{5}{9}\left(w^{\frac{12}{7}}\right)^{-\frac{4}{9}}\times \frac{12}{7}w^{\frac{12}{7}-1}$$

Simplify.

$$\frac{20}{21}w^{\frac{5}{7}}\left(w^{\frac{12}{7}}\right)^{-\frac{4}{9}}$$

Evaluate

$$w^{\frac{20}{21}}$$