Solve 12782x1281

$$12782x1281$$

Evaluate

$$12782x_{1281}$$

Steps For Long Multiplication

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

$$\begin{array}{c}\phantom{\times999}12782\\\underline{\times\phantom{9999}1281}\\\end{array}$$

Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 12782 with 1. Write the result 12782 at the end leaving 0 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999}12782\\\underline{\times\phantom{9999}1281}\\\phantom{\times999}12782\\\end{array}$$

Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 12782 with 8. Write the result 102256 at the end leaving 1 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999}12782\\\underline{\times\phantom{9999}1281}\\\phantom{\times999}12782\\\phantom{\times9}102256\phantom{9}\\\end{array}$$

Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 12782 with 2. Write the result 25564 at the end leaving 2 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999}12782\\\underline{\times\phantom{9999}1281}\\\phantom{\times999}12782\\\phantom{\times9}102256\phantom{9}\\\phantom{\times9}25564\phantom{99}\\\end{array}$$

Now multiply the first number with the 4^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 12782 with 1. Write the result 12782 at the end leaving 3 spaces to the right like this.

$$\begin{array}{c}\phantom{\times999}12782\\\underline{\times\phantom{9999}1281}\\\phantom{\times999}12782\\\phantom{\times9}102256\phantom{9}\\\phantom{\times9}25564\phantom{99}\\\underline{\phantom{\times}12782\phantom{999}}\\\end{array}$$

Now add the intermediate results to get final answer.

$$\begin{array}{c}\phantom{\times999}12782\\\underline{\times\phantom{9999}1281}\\\phantom{\times999}12782\\\phantom{\times9}102256\phantom{9}\\\phantom{\times9}25564\phantom{99}\\\underline{\phantom{\times}12782\phantom{999}}\\\phantom{\times}16373742\end{array}$$

Differentiate w.r.t. x_1281

$$12782$$

Steps Using Definition of a Derivative

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

$$12782x_{1281}^{1-1}$$

Subtract 1 from 1.

$$12782x_{1281}^{0}$$

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

$$12782\times 1$$

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

$$12782$$