Solve (v^6)^frac{3{4}}

$$( v ^ { 6 } ) ^ { \frac { 3 } { 4 } }$$

Differentiate w.r.t. v

$$\frac{9sign(v)\left(|v|\right)^{\frac{7}{2}}}{2}$$

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).

$$\frac{3}{4}\left(v^{6}\right)^{\frac{3}{4}-1}\frac{\mathrm{d}}{\mathrm{d}v}(v^{6})$$

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}.

$$\frac{3}{4}\left(v^{6}\right)^{-\frac{1}{4}}\times 6v^{6-1}$$

Simplify.

$$\frac{9}{2}v^{5}\left(v^{6}\right)^{-\frac{1}{4}}$$

Evaluate

$$\left(|v|\right)^{\frac{9}{2}}$$