Solve frac{1/x^frac{1{2}}}{x^-1}

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

Differentiate w.r.t. x

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

Steps Using Chain Rule

Express \frac{\frac{1}{x^{\frac{1}{2}}}}{x^{-1}} as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{\frac{1}{2}}x^{-1}})$$

To multiply powers of the same base, add their exponents. Add \frac{1}{2} and -1 to get -\frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{1}{x^{-\frac{1}{2}}})$$

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).

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

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}.

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

Simplify.

$$\frac{1}{2}x^{-\frac{3}{2}}\left(x^{-\frac{1}{2}}\right)^{-2}$$

Evaluate

$$\sqrt{x}$$