Solve 2000x5200

$$2000x5200$$

Evaluate

$$2000x_{5200}$$

Steps For Long Multiplication

First line up the numbers vertically and match the places from the right like this.

$$\begin{array}{c}\phantom{\times9999}2000\\\underline{\times\phantom{9999}5200}\\\end{array}$$

Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 2000 with 0. Write the result 0 at the end leaving 0 spaces to the right like this.

$$\begin{array}{c}\phantom{\times9999}2000\\\underline{\times\phantom{9999}5200}\\\phantom{\times99999999}0\\\end{array}$$

Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 2000 with 0. Write the result 0 at the end leaving 1 spaces to the right like this.

$$\begin{array}{c}\phantom{\times9999}2000\\\underline{\times\phantom{9999}5200}\\\phantom{\times99999999}0\\\phantom{\times9999999}0\phantom{9}\\\end{array}$$

Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 2000 with 2. Write the result 4000 at the end leaving 2 spaces to the right like this.

$$\begin{array}{c}\phantom{\times9999}2000\\\underline{\times\phantom{9999}5200}\\\phantom{\times99999999}0\\\phantom{\times9999999}0\phantom{9}\\\phantom{\times99}4000\phantom{99}\\\end{array}$$

Now multiply the first number with the 4^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 2000 with 5. Write the result 10000 at the end leaving 3 spaces to the right like this.

$$\begin{array}{c}\phantom{\times9999}2000\\\underline{\times\phantom{9999}5200}\\\phantom{\times99999999}0\\\phantom{\times9999999}0\phantom{9}\\\phantom{\times99}4000\phantom{99}\\\underline{\phantom{\times}10000\phantom{999}}\\\end{array}$$

Now add the intermediate results to get final answer.

$$\begin{array}{c}\phantom{\times9999}2000\\\underline{\times\phantom{9999}5200}\\\phantom{\times99999999}0\\\phantom{\times9999999}0\phantom{9}\\\phantom{\times99}4000\phantom{99}\\\underline{\phantom{\times}10000\phantom{999}}\\\phantom{\times}10400000\end{array}$$

Differentiate w.r.t. x_5200

$$2000$$

Steps Using Definition of a Derivative

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

$$2000x_{5200}^{1-1}$$

Subtract 1 from 1.

$$2000x_{5200}^{0}$$

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

$$2000\times 1$$

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

$$2000$$