Solve (3/4)(-4z)+5/4z)(-1/2)+1/2z

$$( \frac { 3 } { 4 } ) ( - 4 z ) + \frac { 5 } { 4 } z ) ( - \frac { 1 } { 2 } ) + \frac { 1 } { 2 } z$$

Evaluate

$$-\frac{7z}{4}$$

Solution Steps

Express \frac{3}{4}\left(-4\right) as a single fraction.

$$\frac{3\left(-4\right)}{4}z+\frac{5}{4}z$$

Multiply 3 and -4 to get -12.

$$\frac{-12}{4}z+\frac{5}{4}z$$

Divide -12 by 4 to get -3.

$$-3z+\frac{5}{4}z$$

Combine -3z and \frac{5}{4}z to get -\frac{7}{4}z.

$$-\frac{7}{4}z$$

Differentiate w.r.t. z

$$-\frac{7}{4} = -1\frac{3}{4} = -1.75$$

Steps Using Definition of a Derivative

Express \frac{3}{4}\left(-4\right) as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{3\left(-4\right)}{4}z+\frac{5}{4}z)$$

Multiply 3 and -4 to get -12.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{-12}{4}z+\frac{5}{4}z)$$

Divide -12 by 4 to get -3.

$$\frac{\mathrm{d}}{\mathrm{d}z}(-3z+\frac{5}{4}z)$$

Combine -3z and \frac{5}{4}z to get -\frac{7}{4}z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(-\frac{7}{4}z)$$

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

$$-\frac{7}{4}z^{1-1}$$

Subtract 1 from 1.

$$-\frac{7}{4}z^{0}$$

For any term t except 0, t^{0}=1.

$$-\frac{7}{4}$$