Solve -5z-34+21z^2-45z+83z+z^3-42z^2+5=

$$-5z-34+21 { z }^{ 2 } -45z+83z+ { z }^{ 3 } -42 { z }^{ 2 } +5=$$

Evaluate

$$z^{3}-21z^{2}+33z-29$$

Solution Steps

Combine -5z and -45z to get -50z.

$$-50z-34+21z^{2}+83z+z^{3}-42z^{2}+5$$

Combine -50z and 83z to get 33z.

$$33z-34+21z^{2}+z^{3}-42z^{2}+5$$

Combine 21z^{2} and -42z^{2} to get -21z^{2}.

$$33z-34-21z^{2}+z^{3}+5$$

Add -34 and 5 to get -29.

$$33z-29-21z^{2}+z^{3}$$

Differentiate w.r.t. z

$$3\left(z^{2}-14z+11\right)$$

Steps Using Derivative Rule for Sum

Combine -5z and -45z to get -50z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(-50z-34+21z^{2}+83z+z^{3}-42z^{2}+5)$$

Combine -50z and 83z to get 33z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(33z-34+21z^{2}+z^{3}-42z^{2}+5)$$

Combine 21z^{2} and -42z^{2} to get -21z^{2}.

$$\frac{\mathrm{d}}{\mathrm{d}z}(33z-34-21z^{2}+z^{3}+5)$$

Add -34 and 5 to get -29.

$$\frac{\mathrm{d}}{\mathrm{d}z}(33z-29-21z^{2}+z^{3})$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is 0. The derivative of ax^{n} is nax^{n-1}.

$$33z^{1-1}+2\left(-21\right)z^{2-1}+3z^{3-1}$$

Subtract 1 from 1.

$$33z^{0}+2\left(-21\right)z^{2-1}+3z^{3-1}$$

Multiply 2 times -21.

$$33z^{0}-42z^{2-1}+3z^{3-1}$$

Subtract 1 from 2.

$$33z^{0}-42z^{1}+3z^{3-1}$$

Subtract 1 from 3.

$$33z^{0}-42z^{1}+3z^{2}$$

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

$$33z^{0}-42z+3z^{2}$$

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

$$33\times 1-42z+3z^{2}$$

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

$$33-42z+3z^{2}$$