Calculate the derivative at a point using the definition of limit

Hi Bildanur, let's solve this calculus problem together. We are asked to evaluate the limit of f of x minus f of two over x minus two, as x approaches two.

First, let's recognize that this limit expression is actually the definition of the derivative of the function at the point where x equals two.

So, instead of computing the limit directly, we can find the derivative f prime of x and then substitute two into it.

Let's perform the differentiation. We use the power rule, which says that the derivative of x to the power of n is n times x to the power of n minus one.

For the first term, four x cubed, the derivative is four times three, which is twelve, times x squared.

Next, the derivative of five x squared is five times two, which is ten, times x.