Solve 1/2a^5:1/4a^3

$$\frac { 1 } { 2 } a ^ { 5 } : \frac { 1 } { 4 } a ^ { 3 }$$

Evaluate

$$2a^{8}$$

Solution Steps

Divide \frac{1}{2}a^{5} by \frac{1}{4} by multiplying \frac{1}{2}a^{5} by the reciprocal of \frac{1}{4}.

$$\frac{1}{2}a^{5}\times 4a^{3}$$

Multiply \frac{1}{2} and 4 to get 2.

$$2a^{5}a^{3}$$

To multiply powers of the same base, add their exponents. Add 5 and 3 to get 8.

$$2a^{8}$$

Differentiate w.r.t. a

$$16a^{7}$$

Steps Using Definition of a Derivative

Divide \frac{1}{2}a^{5} by \frac{1}{4} by multiplying \frac{1}{2}a^{5} by the reciprocal of \frac{1}{4}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{2}a^{5}\times 4a^{3})$$

Multiply \frac{1}{2} and 4 to get 2.

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

To multiply powers of the same base, add their exponents. Add 5 and 3 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(2a^{8})$$

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

$$8\times 2a^{8-1}$$

Multiply 8 times 2.

$$16a^{8-1}$$

Subtract 1 from 8.

$$16a^{7}$$