Solve 1/8x^-frac{1{2}}div(3/5*10/9)]

$$\frac { 1 } { 8 } x ^ { - \frac { 1 } { 2 } } \div ( \frac { 3 } { 5 } \times \frac { 10 } { 9 } ) ]$$

Evaluate

$$\frac{3}{16\sqrt{x}}$$

Solution Steps

Multiply \frac{3}{5} and \frac{10}{9} to get \frac{2}{3}.

$$\frac{\frac{1}{8}x^{-\frac{1}{2}}}{\frac{2}{3}}$$

Divide \frac{1}{8}x^{-\frac{1}{2}} by \frac{2}{3} by multiplying \frac{1}{8}x^{-\frac{1}{2}} by the reciprocal of \frac{2}{3}.

$$\frac{\frac{1}{8}x^{-\frac{1}{2}}\times 3}{2}$$

Multiply \frac{1}{8} and 3 to get \frac{3}{8}.

$$\frac{\frac{3}{8}x^{-\frac{1}{2}}}{2}$$

Divide \frac{3}{8}x^{-\frac{1}{2}} by 2 to get \frac{3}{16}x^{-\frac{1}{2}}.

$$\frac{3}{16}x^{-\frac{1}{2}}$$

Differentiate w.r.t. x

$$-\frac{3}{32x^{\frac{3}{2}}}$$

Steps Using Definition of a Derivative

Multiply \frac{3}{5} and \frac{10}{9} to get \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{8}x^{-\frac{1}{2}}}{\frac{2}{3}})$$

Divide \frac{1}{8}x^{-\frac{1}{2}} by \frac{2}{3} by multiplying \frac{1}{8}x^{-\frac{1}{2}} by the reciprocal of \frac{2}{3}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{1}{8}x^{-\frac{1}{2}}\times 3}{2})$$

Multiply \frac{1}{8} and 3 to get \frac{3}{8}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{\frac{3}{8}x^{-\frac{1}{2}}}{2})$$

Divide \frac{3}{8}x^{-\frac{1}{2}} by 2 to get \frac{3}{16}x^{-\frac{1}{2}}.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{3}{16}x^{-\frac{1}{2}})$$

The derivative of ax^{n} is nax^{n-1}.

$$-\frac{1}{2}\times \frac{3}{16}x^{-\frac{1}{2}-1}$$

Multiply -\frac{1}{2} times \frac{3}{16} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$-\frac{3}{32}x^{-\frac{1}{2}-1}$$

Subtract 1 from -\frac{1}{2}.

$$-\frac{3}{32}x^{-\frac{3}{2}}$$