Step 1

The function that we are going to find the derivative of by using the difference quotient or the definition of the derivative is;
f(x)= x Â²-5x+6.Please click on the image to see how we use the difference quotient to find the derivative since it is difficult to demonstrate the process in this step.

Step 2

We will substitute the expression (x+Ã¢ÂˆÂ†x) into the function
f(x)= x Â²-5x+6 so that we have f(x+Ã¢ÂˆÂ†x)= (x+Ã¢ÂˆÂ†x) Â²-5(x+Ã¢ÂˆÂ†x)+6 which is equal to x Â²+2xÃ¢ÂˆÂ†x+(Ã¢ÂˆÂ†x) Â²-5x-5Ã¢ÂˆÂ†x+6. Now, we know that f(x)=x Â²-5x+6 we will now subtract f(x) from f(x+Ã¢ÂˆÂ†x) which is equal to
x Â²+2xÃ¢ÂˆÂ†x+(Ã¢ÂˆÂ†x) Â²-5x-5Ã¢ÂˆÂ†x+6-(x Â²-5x+6) = x Â²+2xÃ¢ÂˆÂ†x+Ã¢ÂˆÂ†x Â²-5x-5Ã¢ÂˆÂ†x+6-x Â²+5x-6
= 2xÃ¢ÂˆÂ†x+(Ã¢ÂˆÂ†x) Â²-5Ã¢ÂˆÂ†x. Please click on the image for better understanding.

Step 3

Now we will find the quotient of 2xÃ¢ÂˆÂ†x+(Ã¢ÂˆÂ†x) Â²-5Ã¢ÂˆÂ†x with Ã¢ÂˆÂ†x. That is,by factoring out the Ã¢ÂˆÂ†x and dividing by Ã¢ÂˆÂ†x, since Ã¢ÂˆÂ†x approaching 0 is not equal to 0 therefore we can divide by Ã¢ÂˆÂ†x. We have(2xÃ¢ÂˆÂ†x+(Ã¢ÂˆÂ†x) Â²-5Ã¢ÂˆÂ†x)/Ã¢ÂˆÂ†x which is equal to 2x+Ã¢ÂˆÂ†x-5. Please click on the image for better understanding.

Step 4

Finally, the limit of 2x+Ã¢ÂˆÂ†x-5, as Ã¢ÂˆÂ†x approaches 0 is equal to 2x-5.
Hence, the derivative of f(x)= x Â²-5x+6 is equal to f'(x)= 2x-5.