Solve (64x^6)^frac{3{2}}

$$( 64 x ^ { 6 } ) ^ { \frac { 3 } { 2 } }$$

Evaluate

$$512\left(|x|\right)^{9}$$

Solution Steps

Expand \left(64x^{6}\right)^{\frac{3}{2}}.

$$64^{\frac{3}{2}}\left(x^{6}\right)^{\frac{3}{2}}$$

To raise a power to another power, multiply the exponents. Multiply 6 and \frac{3}{2} to get 9.

$$64^{\frac{3}{2}}x^{9}$$

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

$$512x^{9}$$

Differentiate w.r.t. x

$$4608|x|x^{7}$$

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).

$$\frac{3}{2}\times \left(64x^{6}\right)^{\frac{3}{2}-1}\frac{\mathrm{d}}{\mathrm{d}x}(64x^{6})$$

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}.

$$\frac{3}{2}\sqrt{64x^{6}}\times 6\times 64x^{6-1}$$

Simplify.

$$576x^{5}\sqrt{64x^{6}}$$