Solve 17/5a^{3}*(-5/34a^{2})]^{2}:(-1/2a^3):(-3/4a^3)*(-frac{2}{3}b^2)^2=

$$\frac { 17 } { 5 } a ^ { 3 } \cdot ( - \frac { 5 } { 34 } a ^ { 2 } ) ] ^ { 2 } : ( - \frac { 1 } { 2 } a ^ { 3 } ) : ( - \frac { 3 } { 4 } a ^ { 3 } ) \cdot ( - \frac { 2 } { 3 } b ^ { 2 } ) ^ { 2 } =$$

Evaluate

$$-\frac{a^{5}}{2}$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{17}{5}a^{5}\left(-\frac{5}{34}\right)$$

Multiply \frac{17}{5} and -\frac{5}{34} to get -\frac{1}{2}.

$$-\frac{1}{2}a^{5}$$

Differentiate w.r.t. a

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

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 3 and 2 to get 5.

$$\frac{\mathrm{d}}{\mathrm{d}a}(\frac{17}{5}a^{5}\left(-\frac{5}{34}\right))$$

Multiply \frac{17}{5} and -\frac{5}{34} to get -\frac{1}{2}.

$$\frac{\mathrm{d}}{\mathrm{d}a}(-\frac{1}{2}a^{5})$$

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

$$5\left(-\frac{1}{2}\right)a^{5-1}$$

Multiply 5 times -\frac{1}{2}.

$$-\frac{5}{2}a^{5-1}$$

Subtract 1 from 5.

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