Solve 10,000^frac{1{4}}*z+10,000^frac{1{2}}*z

$$10,000 ^ { \frac { 1 } { 4 } } \cdot z + 10,000 ^ { \frac { 1 } { 2 } } \cdot z$$

Evaluate

$$110z$$

Solution Steps

Calculate 10000 to the power of \frac{1}{4} and get 10.

$$10z+10000^{\frac{1}{2}}z$$

Calculate 10000 to the power of \frac{1}{2} and get 100.

$$10z+100z$$

Combine 10z and 100z to get 110z.

$$110z$$

Differentiate w.r.t. z

$$110$$

Steps Using Definition of a Derivative

Calculate 10000 to the power of \frac{1}{4} and get 10.

$$\frac{\mathrm{d}}{\mathrm{d}z}(10z+10000^{\frac{1}{2}}z)$$

Calculate 10000 to the power of \frac{1}{2} and get 100.

$$\frac{\mathrm{d}}{\mathrm{d}z}(10z+100z)$$

Combine 10z and 100z to get 110z.

$$\frac{\mathrm{d}}{\mathrm{d}z}(110z)$$

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

$$110z^{1-1}$$

Subtract 1 from 1.

$$110z^{0}$$

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

$$110\times 1$$

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

$$110$$