Solve 4b^-3d^2/8b^2d^-5

$$\frac { 4 b ^ { - 3 } d ^ { 2 } } { 8 b ^ { 2 } d ^ { - 5 } }$$

Evaluate

$$\frac{d^{7}}{2b^{5}}$$

Solution Steps

Cancel out 4 in both numerator and denominator.

$$\frac{b^{-3}d^{2}}{2d^{-5}b^{2}}$$

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

$$\frac{b^{-3}d^{7}}{2b^{2}}$$

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

$$\frac{d^{7}}{2b^{5}}$$

Differentiate w.r.t. b

$$-\frac{5d^{7}}{2b^{6}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{4d^{2}d^{5}}{8}b^{-3-2})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}b}(\frac{d^{7}}{2}b^{-5})$$

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}.

$$-5\times \frac{d^{7}}{2}b^{-5-1}$$

Do the arithmetic.

$$\left(-\frac{5d^{7}}{2}\right)b^{-6}$$