Solve (y^8)^frac{4{5}}

$$( y ^ { 8 } ) ^ { \frac { 4 } { 5 } }$$

Differentiate w.r.t. y

$$\frac{32y^{\frac{27}{5}}}{5}$$

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).

$$\frac{4}{5}\left(y^{8}\right)^{\frac{4}{5}-1}\frac{\mathrm{d}}{\mathrm{d}y}(y^{8})$$

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}.

$$\frac{4}{5}\left(y^{8}\right)^{-\frac{1}{5}}\times 8y^{8-1}$$

Simplify.

$$\frac{32}{5}y^{7}\left(y^{8}\right)^{-\frac{1}{5}}$$

Evaluate

$$y^{\frac{32}{5}}$$