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.
We will substitute the expression (x+Ã¢ÂÂx) into the function
f(x)= x Â²-5x+6 so that we have f(x+Ã¢ÂÂx)= (x+Ã¢ÂÂx) Â²-5(x+Ã¢ÂÂx)+6 which is equal to x Â²+2xÃ¢ÂÂx+(Ã¢ÂÂx) Â²-5x-5Ã¢ÂÂx+6. Now, we know that f(x)=x Â²-5x+6 we will now subtract f(x) from f(x+Ã¢ÂÂx) which is equal to
x Â²+2xÃ¢ÂÂx+(Ã¢ÂÂx) Â²-5x-5Ã¢ÂÂx+6-(x Â²-5x+6) = x Â²+2xÃ¢ÂÂx+Ã¢ÂÂx Â²-5x-5Ã¢ÂÂx+6-x Â²+5x-6
= 2xÃ¢ÂÂx+(Ã¢ÂÂx) Â²-5Ã¢ÂÂx. Please click on the image for better understanding.
Now we will find the quotient of 2xÃ¢ÂÂx+(Ã¢ÂÂx) Â²-5Ã¢ÂÂx with Ã¢ÂÂx. That is,by factoring out the Ã¢ÂÂx and dividing by Ã¢ÂÂx, since Ã¢ÂÂx approaching 0 is not equal to 0 therefore we can divide by Ã¢ÂÂx. We have(2xÃ¢ÂÂx+(Ã¢ÂÂx) Â²-5Ã¢ÂÂx)/Ã¢ÂÂx which is equal to 2x+Ã¢ÂÂx-5. Please click on the image for better understanding.
Finally, the limit of 2x+Ã¢ÂÂx-5, as Ã¢ÂÂx approaches 0 is equal to 2x-5.
Hence, the derivative of f(x)= x Â²-5x+6 is equal to f'(x)= 2x-5.