Solve (64x^4)^frac{5{6}}

$$( 64 x ^ { 4 } ) ^ { \frac { 5 } { 6 } }$$

Evaluate

$$32x^{\frac{10}{3}}$$

Solution Steps

Expand \left(64x^{4}\right)^{\frac{5}{6}}.

$$64^{\frac{5}{6}}\left(x^{4}\right)^{\frac{5}{6}}$$

To raise a power to another power, multiply the exponents. Multiply 4 and \frac{5}{6} to get \frac{10}{3}.

$$64^{\frac{5}{6}}x^{\frac{10}{3}}$$

Calculate 64 to the power of \frac{5}{6} and get 32.

$$32x^{\frac{10}{3}}$$

Differentiate w.r.t. x

$$\frac{320x^{\frac{7}{3}}}{3}$$

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).

$$\frac{5}{6}\times \left(64x^{4}\right)^{\frac{5}{6}-1}\frac{\mathrm{d}}{\mathrm{d}x}(64x^{4})$$

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}.

$$\frac{5}{6}\times \left(64x^{4}\right)^{-\frac{1}{6}}\times 4\times 64x^{4-1}$$

Simplify.

$$\frac{640}{3}x^{3}\times \left(64x^{4}\right)^{-\frac{1}{6}}$$