Solve 1/3a^5*(-2/5a^2)*(-3/4a)

$$\frac { 1 } { 3 } a ^ { 5 } \cdot ( \frac { - 2 } { 5 } a ^ { 2 } ) \cdot ( \frac { - 3 } { 4 } a )$$

Evaluate

$$\frac{a^{8}}{10}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

$$\frac{1}{3}a^{7}\times \frac{-2}{5}\times \frac{-3}{4}a$$

To multiply powers of the same base, add their exponents. Add 7 and 1 to get 8.

$$\frac{1}{3}a^{8}\times \frac{-2}{5}\times \frac{-3}{4}$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$\frac{1}{3}a^{8}\left(-\frac{2}{5}\right)\times \frac{-3}{4}$$

Multiply \frac{1}{3} and -\frac{2}{5} to get -\frac{2}{15}.

$$-\frac{2}{15}a^{8}\times \frac{-3}{4}$$

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$-\frac{2}{15}a^{8}\left(-\frac{3}{4}\right)$$

Multiply -\frac{2}{15} and -\frac{3}{4} to get \frac{1}{10}.

$$\frac{1}{10}a^{8}$$

Differentiate w.r.t. a

$$\frac{4a^{7}}{5}$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 5 and 2 to get 7.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{3}a^{7}\times \frac{-2}{5}\times \frac{-3}{4}a)$$

To multiply powers of the same base, add their exponents. Add 7 and 1 to get 8.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{3}a^{8}\times \frac{-2}{5}\times \frac{-3}{4})$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{3}a^{8}\left(-\frac{2}{5}\right)\times \frac{-3}{4})$$

Multiply \frac{1}{3} and -\frac{2}{5} to get -\frac{2}{15}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{2}{15}a^{8}\times \frac{-3}{4})$$

Fraction \frac{-3}{4} can be rewritten as -\frac{3}{4} by extracting the negative sign.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{2}{15}a^{8}\left(-\frac{3}{4}\right))$$

Multiply -\frac{2}{15} and -\frac{3}{4} to get \frac{1}{10}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{1}{10}a^{8})$$

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

$$8\times \frac{1}{10}a^{8-1}$$

Multiply 8 times \frac{1}{10}.

$$\frac{4}{5}a^{8-1}$$

Subtract 1 from 8.

$$\frac{4}{5}a^{7}$$