Solve P_{0}e^-000012t

$$P _ { 0 } e ^ { - 000012 t }$$

Evaluate

$$P_{0}$$

Solution Steps

Multiply 0 and 0 to get 0.

$$P_{0}e^{0\times 0\times 0\times 12t}$$

Multiply 0 and 0 to get 0.

$$P_{0}e^{0\times 0\times 12t}$$

Multiply 0 and 0 to get 0.

$$P_{0}e^{0\times 12t}$$

Multiply 0 and 12 to get 0.

$$P_{0}e^{0t}$$

Anything times zero gives zero.

$$P_{0}e^{0}$$

Calculate e to the power of 0 and get 1.

$$P_{0}\times 1$$

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

$$P_{0}$$

Differentiate w.r.t. P_0

$$1$$

Steps Using Definition of a Derivative

Multiply 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}P_{0}}(P_{0}e^{0\times 0\times 0\times 12t})$$

Multiply 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}P_{0}}(P_{0}e^{0\times 0\times 12t})$$

Multiply 0 and 0 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}P_{0}}(P_{0}e^{0\times 12t})$$

Multiply 0 and 12 to get 0.

$$\frac{\mathrm{d}}{\mathrm{d}P_{0}}(P_{0}e^{0t})$$

Anything times zero gives zero.

$$\frac{\mathrm{d}}{\mathrm{d}P_{0}}(P_{0}e^{0})$$

Calculate e to the power of 0 and get 1.

$$\frac{\mathrm{d}}{\mathrm{d}P_{0}}(P_{0}\times 1)$$

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

$$P_{0}^{1-1}$$

Subtract 1 from 1.

$$P_{0}^{0}$$

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

$$1$$