Solve 5ab^-5/5^-1a^-6b^3

$$\frac { 5 a b ^ { - 5 } } { 5 ^ { - 1 } a ^ { - 6 } b ^ { 3 } }$$

Evaluate

$$\frac{25a^{7}}{b^{8}}$$

Solution Steps

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

$$\frac{5^{2}b^{-5}a^{7}}{b^{3}}$$

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

$$\frac{5^{2}a^{7}}{b^{8}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{25a^{7}}{b^{8}}$$

Differentiate w.r.t. a

$$\frac{175a^{6}}{b^{8}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5}{b^{5}\times \frac{b^{3}}{5}}a^{1-\left(-6\right)})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{25}{b^{8}}a^{7})$$

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}.

$$7\times \frac{25}{b^{8}}a^{7-1}$$

Do the arithmetic.

$$\frac{175}{b^{8}}a^{6}$$