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 ²-5x+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.

Difference Quotient Formula

Step 2

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

Step 3

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

Step 4

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