Solve (16x^-16)^frac{1{4}}

$$( 16 x ^ { - 16 } ) ^ { \frac { 1 } { 4 } }$$

Evaluate

$$\frac{2}{x^{4}}$$

Solution Steps

Expand \left(16x^{-16}\right)^{\frac{1}{4}}.

$$16^{\frac{1}{4}}\left(x^{-16}\right)^{\frac{1}{4}}$$

To raise a power to another power, multiply the exponents. Multiply -16 and \frac{1}{4} to get -4.

$$16^{\frac{1}{4}}x^{-4}$$

Calculate 16 to the power of \frac{1}{4} and get 2.

$$2x^{-4}$$

Differentiate w.r.t. x

$$-\frac{8}{x^{5}}$$

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).

$$\frac{1}{4}\times \left(16x^{-16}\right)^{\frac{1}{4}-1}\frac{\mathrm{d}}{\mathrm{d}x}(16x^{-16})$$

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}.

$$\frac{1}{4}\times \left(16x^{-16}\right)^{-\frac{3}{4}}\left(-16\right)\times 16x^{-16-1}$$

Simplify.

$$-64x^{-17}\times \left(16x^{-16}\right)^{-\frac{3}{4}}$$