Solve (1/4y^3)^-frac{1{2}}

$$( \frac { 1 } { 4 } y ^ { 3 } ) ^ { - \frac { 1 } { 2 } }$$

Differentiate w.r.t. y

$$-\frac{3}{y^{\frac{5}{2}}}$$

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).

$$-\frac{1}{2}\times \left(\frac{1}{4}y^{3}\right)^{-\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}y}(\frac{1}{4}y^{3})$$

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}.

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

Simplify.

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

Evaluate

$$\frac{2}{y^{\frac{3}{2}}}$$