Solve (frac{6f^{2}g^-8h^2}{32fg^-5h^4})

$$( \frac { 6 f ^ { 2 } g ^ { - 8 } h ^ { 2 } } { 32 f g ^ { - 5 } h ^ { 4 } } )$$

Evaluate

$$\frac{3f}{16h^{2}g^{3}}$$

Solution Steps

Cancel out 2fh^{2} in both numerator and denominator.

$$\frac{3g^{-8}f}{16g^{-5}h^{2}}$$

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

$$\frac{3f}{16h^{2}g^{3}}$$

Differentiate w.r.t. f

$$\frac{3}{16h^{2}g^{3}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{6h^{2}}{g^{8}\times \frac{32h^{4}}{g^{5}}}f^{2-1})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}f}(\frac{3}{16h^{2}g^{3}}f^{1})$$

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}.

$$\frac{3}{16h^{2}g^{3}}f^{1-1}$$

Do the arithmetic.

$$\frac{3}{16h^{2}g^{3}}f^{0}$$

For any term t except 0, t^{0}=1.

$$\frac{3}{16h^{2}g^{3}}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{3}{16h^{2}g^{3}}$$