Find the Derivative of a function by using the difference quotient definition

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zadocksmath | Sep 4, 2009 |

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_{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.

Step 2

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

Step 3

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

Step 4

Finally, the limit of 2x+âˆ†x-5, as âˆ†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.