Solve 5a^-7b^4/20a^3b^-2

$$\frac { 5 a ^ { - 7 } b ^ { 4 } } { 20 a ^ { 3 } b ^ { - 2 } }$$

Evaluate

$$\frac{b^{6}}{4a^{10}}$$

Solution Steps

Cancel out 5 in both numerator and denominator.

$$\frac{a^{-7}b^{4}}{4b^{-2}a^{3}}$$

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

$$\frac{a^{-7}b^{6}}{4a^{3}}$$

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

$$\frac{b^{6}}{4a^{10}}$$

Differentiate w.r.t. a

$$-\frac{5b^{6}}{2a^{11}}$$

Solution Steps

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{5b^{4}b^{2}}{20}a^{-7-3})$$

Do the arithmetic.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{b^{6}}{4}a^{-10})$$

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}.

$$-10\times \frac{b^{6}}{4}a^{-10-1}$$

Do the arithmetic.

$$\left(-\frac{5b^{6}}{2}\right)a^{-11}$$