Solve (-8/43y^3/4)^-1/3

$$( - \frac { 8 } { 43 } y ^ { 3 / 4 } ) ^ { - 1 / 3 }$$

Differentiate w.r.t. y

$$\frac{\sqrt[3]{43}}{8y^{\frac{5}{4}}}$$

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).

$$-\frac{1}{3}\left(-\frac{8}{43}y^{\frac{3}{4}}\right)^{-\frac{1}{3}-1}\frac{\mathrm{d}}{\mathrm{d}y}(-\frac{8}{43}y^{\frac{3}{4}})$$

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}.

$$-\frac{1}{3}\left(-\frac{8}{43}y^{\frac{3}{4}}\right)^{-\frac{4}{3}}\times \frac{3}{4}\left(-\frac{8}{43}\right)y^{\frac{3}{4}-1}$$

Simplify.

$$\frac{2}{43}y^{-\frac{1}{4}}\left(-\frac{8}{43}y^{\frac{3}{4}}\right)^{-\frac{4}{3}}$$

Evaluate

$$-\frac{\sqrt[3]{43}}{2\sqrt[4]{y}}$$