Solve (64h^4)^frac{3{2}}

$$( 64 h ^ { 4 } ) ^ { \frac { 3 } { 2 } }$$

Evaluate

$$512h^{6}$$

Solution Steps

Expand \left(64h^{4}\right)^{\frac{3}{2}}.

$$64^{\frac{3}{2}}\left(h^{4}\right)^{\frac{3}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and \frac{3}{2} to get 6.

$$64^{\frac{3}{2}}h^{6}$$

Calculate 64 to the power of \frac{3}{2} and get 512.

$$512h^{6}$$

Differentiate w.r.t. h

$$3072h^{5}$$

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).

$$\frac{3}{2}\times \left(64h^{4}\right)^{\frac{3}{2}-1}\frac{\mathrm{d}}{\mathrm{d}h}(64h^{4})$$

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}.

$$\frac{3}{2}\sqrt{64h^{4}}\times 4\times 64h^{4-1}$$

Simplify.

$$384h^{3}\sqrt{64h^{4}}$$