Solve 5a^-2b^3/25a^5b^2

$$\frac { 5 a ^ { - 2 } b ^ { 3 } } { 25 a ^ { 5 } b ^ { 2 } }$$

Evaluate

$$\frac{b}{5a^{7}}$$

Solution Steps

Cancel out 5b^{2} in both numerator and denominator.

$$\frac{a^{-2}b}{5a^{5}}$$

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

$$\frac{b}{5a^{7}}$$

Differentiate w.r.t. a

$$-\frac{7b}{5a^{8}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5b^{3}}{25b^{2}}a^{-2-5})$$

Do the arithmetic.

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

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}.

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

Do the arithmetic.

$$\left(-\frac{7b}{5}\right)a^{-8}$$