How do you find the derivative of #f(x)=x^2-5x+3#?
The derivative of a function is the instantaneous rate of change of the function at a given point. But we have a function, so we want to find the rate of change throughout. We have two ways of doing this, and I'll show you both ways.
Method 1: The Power Rule
This method only works for polynomials (
So we can apply this to your function. But we also have a constant. Remember that adding or subtracting a constant yields only a vertical shift, not a stretch. Therefore, it has no effect on the rate of change, so we can also say:
We can use these two rules together to get the answer:
Method 2: First Principles
First Principles is a method that allows you to take the derivative of any function. It comes from the slope formula (
So let's plug in your function:
Since we can't divide by 0, we need to get the h out of the denominator. So let's expand the binomial.
And let's combine all the like terms (cancelling out most of the x values):
Now, every term as an h somewhere in it. So we can divide by h:
And now as the limit approaches 0, 2h reaches 0, and therefore, we are left with the final answer: