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

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

Evaluate

$$\frac{a^{6}}{49}$$

Solution Steps

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

$$\frac{7^{-5}a^{6}}{7^{-3}}$$

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

$$\frac{a^{6}}{7^{2}}$$

Calculate 7 to the power of 2 and get 49.

$$\frac{a^{6}}{49}$$

Differentiate w.r.t. a

$$\frac{6a^{5}}{49}$$

Steps Using Definition of a Derivative

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{7^{-5}a^{6}}{7^{-3}})$$

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

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{6}}{7^{2}})$$

Calculate 7 to the power of 2 and get 49.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{a^{6}}{49})$$

The derivative of ax^{n} is nax^{n-1}.

$$6\times \frac{1}{49}a^{6-1}$$

Multiply 6 times \frac{1}{49}.

$$\frac{6}{49}a^{6-1}$$

Subtract 1 from 6.

$$\frac{6}{49}a^{5}$$