Solve (16x^4)^-1/4

$$( 16 x ^ { 4 } ) ^ { - 1 / 4 }$$

Evaluate

$$\frac{1}{2|x|}$$

Solution Steps

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

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

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

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

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

$$\frac{1}{2}x^{-1}$$

Differentiate w.r.t. x

$$-\frac{|x|}{2x^{3}}$$

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).

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

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}.

$$-\frac{1}{4}\times \left(16x^{4}\right)^{-\frac{5}{4}}\times 4\times 16x^{4-1}$$

Simplify.

$$-16x^{3}\times \left(16x^{4}\right)^{-\frac{5}{4}}$$