Solve [2z-(-4/3z)+(3/2z)]-(-3/4x)-(-1/3z)+(-0,5x-frac{1}{4}x)

$$[ 2 z - ( - \frac { 4 } { 3 } z ) + ( \frac { 3 } { 2 } z ) ] - ( - \frac { 3 } { 4 } x ) - ( - \frac { 1 } { 3 } z ) + ( - 0,5 x - \frac { 1 } { 4 } x )$$

Evaluate

$$\frac{31z}{6}$$

Solution Steps

The opposite of -\frac{4}{3}z is \frac{4}{3}z.

$$2z+\frac{4}{3}z+\frac{3}{2}z-\left(-\frac{3}{4}x\right)-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x$$

Combine 2z and \frac{4}{3}z to get \frac{10}{3}z.

$$\frac{10}{3}z+\frac{3}{2}z-\left(-\frac{3}{4}x\right)-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x$$

Combine \frac{10}{3}z and \frac{3}{2}z to get \frac{29}{6}z.

$$\frac{29}{6}z-\left(-\frac{3}{4}x\right)-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x$$

The opposite of -\frac{3}{4}x is \frac{3}{4}x.

$$\frac{29}{6}z+\frac{3}{4}x-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x$$

The opposite of -\frac{1}{3}z is \frac{1}{3}z.

$$\frac{29}{6}z+\frac{3}{4}x+\frac{1}{3}z-0,5x-\frac{1}{4}x$$

Combine \frac{29}{6}z and \frac{1}{3}z to get \frac{31}{6}z.

$$\frac{31}{6}z+\frac{3}{4}x-0,5x-\frac{1}{4}x$$

Combine \frac{3}{4}x and -0,5x to get \frac{1}{4}x.

$$\frac{31}{6}z+\frac{1}{4}x-\frac{1}{4}x$$

Combine \frac{1}{4}x and -\frac{1}{4}x to get 0.

$$\frac{31}{6}z$$

Differentiate w.r.t. z

$$\frac{31}{6} = 5\frac{1}{6} = 5.166666666666667$$

Steps Using Definition of a Derivative

The opposite of -\frac{4}{3}z is \frac{4}{3}z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(2z+\frac{4}{3}z+\frac{3}{2}z-\left(-\frac{3}{4}x\right)-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x)$$

Combine 2z and \frac{4}{3}z to get \frac{10}{3}z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{10}{3}z+\frac{3}{2}z-\left(-\frac{3}{4}x\right)-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x)$$

Combine \frac{10}{3}z and \frac{3}{2}z to get \frac{29}{6}z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{29}{6}z-\left(-\frac{3}{4}x\right)-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x)$$

The opposite of -\frac{3}{4}x is \frac{3}{4}x.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{29}{6}z+\frac{3}{4}x-\left(-\frac{1}{3}z\right)-0,5x-\frac{1}{4}x)$$

The opposite of -\frac{1}{3}z is \frac{1}{3}z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{29}{6}z+\frac{3}{4}x+\frac{1}{3}z-0,5x-\frac{1}{4}x)$$

Combine \frac{29}{6}z and \frac{1}{3}z to get \frac{31}{6}z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{31}{6}z+\frac{3}{4}x-0,5x-\frac{1}{4}x)$$

Combine \frac{3}{4}x and -0,5x to get \frac{1}{4}x.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{31}{6}z+\frac{1}{4}x-\frac{1}{4}x)$$

Combine \frac{1}{4}x and -\frac{1}{4}x to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}z}(\frac{31}{6}z)$$

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

$$\frac{31}{6}z^{1-1}$$

Subtract 1 from 1.

$$\frac{31}{6}z^{0}$$

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

$$\frac{31}{6}\times 1$$

For any term t, t\times 1=t and 1t=t.

$$\frac{31}{6}$$