Solve Iquad5^2*10^8*5^4

$$I \quad 5 ^ { 2 } \cdot 10 ^ { 8 } \cdot 5 ^ { 4 }$$

Evaluate

$$1562500000000I$$

Solution Steps

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$I\times 5^{6}\times 10^{8}$$

Calculate 5 to the power of 6 and get 15625.

$$I\times 15625\times 10^{8}$$

Calculate 10 to the power of 8 and get 100000000.

$$I\times 15625\times 100000000$$

Multiply 15625 and 100000000 to get 1562500000000.

$$I\times 1562500000000$$

Differentiate w.r.t. I

$$1562500000000$$

Steps Using Definition of a Derivative

To multiply powers of the same base, add their exponents. Add 2 and 4 to get 6.

$$\frac{\mathrm{d}}{\mathrm{d}I}(I\times 5^{6}\times 10^{8})$$

Calculate 5 to the power of 6 and get 15625.

$$\frac{\mathrm{d}}{\mathrm{d}I}(I\times 15625\times 10^{8})$$

Calculate 10 to the power of 8 and get 100000000.

$$\frac{\mathrm{d}}{\mathrm{d}I}(I\times 15625\times 100000000)$$

Multiply 15625 and 100000000 to get 1562500000000.

$$\frac{\mathrm{d}}{\mathrm{d}I}(I\times 1562500000000)$$

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

$$1562500000000I^{1-1}$$

Subtract 1 from 1.

$$1562500000000I^{0}$$

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

$$1562500000000\times 1$$

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

$$1562500000000$$