Solve (25x^-4)^1/2

$$( 25 x ^ { - 4 } ) ^ { 1 / 2 }$$

Evaluate

$$\frac{5}{x^{2}}$$

Solution Steps

Expand \left(25x^{-4}\right)^{\frac{1}{2}}.

$$25^{\frac{1}{2}}\left(x^{-4}\right)^{\frac{1}{2}}$$

To raise a power to another power, multiply the exponents. Multiply -4 and \frac{1}{2} to get -2.

$$25^{\frac{1}{2}}x^{-2}$$

Calculate 25 to the power of \frac{1}{2} and get 5.

$$5x^{-2}$$

Differentiate w.r.t. x

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

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).

$$\frac{1}{2}\times \left(25x^{-4}\right)^{\frac{1}{2}-1}\frac{\mathrm{d}}{\mathrm{d}x}(25x^{-4})$$

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}.

$$\frac{1}{2}\times \left(25x^{-4}\right)^{-\frac{1}{2}}\left(-4\right)\times 25x^{-4-1}$$

Simplify.

$$-50x^{-5}\times \left(25x^{-4}\right)^{-\frac{1}{2}}$$