Solve frac{2m^frac{-3{4}}d^frac{3{4}}}{8m^frac{-11{4}}d^frac{7{4}}}

$$\frac { 2 m ^ { \frac { - 3 } { 4 } } d ^ { \frac { 3 } { 4 } } } { 8 m ^ { \frac { - 11 } { 4 } } d ^ { \frac { 7 } { 4 } } }$$

Evaluate

$$\frac{m^{2}}{4d}$$

Solution Steps

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$\frac{2m^{-\frac{3}{4}}d^{\frac{3}{4}}}{8m^{\frac{-11}{4}}d^{\frac{7}{4}}}$$

Fraction \frac{-11}{4} can be rewritten as -\frac{11}{4} by extracting the negative sign.

$$\frac{2m^{-\frac{3}{4}}d^{\frac{3}{4}}}{8m^{-\frac{11}{4}}d^{\frac{7}{4}}}$$

Cancel out 2d^{\frac{3}{4}} in both numerator and denominator.

$$\frac{m^{-\frac{3}{4}}}{4m^{-\frac{11}{4}}d}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{m^{2}}{4d}$$

Differentiate w.r.t. m

$$\frac{m}{2d}$$

Solution Steps

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{2d^{\frac{3}{4}}}{8d^{\frac{7}{4}}}m^{-\frac{3}{4}-\left(-\frac{11}{4}\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}m}(\frac{1}{4d}m^{2})$$

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}.

$$2\times \frac{1}{4d}m^{2-1}$$

Do the arithmetic.

$$\frac{1}{2d}m^{1}$$

For any term t, t^{1}=t.

$$\frac{1}{2d}m$$