17 A level Mathematics Calculus Differentiating Expressions with Powers of x Part 1 Video 10

Hi everyone, I'm Christina and welcome to video 10 in our A Level Mathematics Calculus series. In this video we're differentiating expressions with powers of X and this is part one. So our objectives in this video are just to introduce formally for differentiating powers of X and we have two special cases. The main one we're going to look at here is to recognize that the derivative of the constant function, that's a function where f of X is equal to a constant number A. When we find the derivative of a constant function we always get zero. So let's have a look at the formulae here. So if n is any real number and A is a constant number, for f of X in the form f of X equals X to the power of n. If we want to find the derivative of that function f dash of X, we bring down the n. So we bring down our power n in front, which becomes our coefficient for our derivative. And then we minus one from the index, which we have here. So that's the general formula we follow when we have a function in the form X to the n. Now in our second formula here, we have a function which is A times X to the n. So we have a coefficient of A in front of our X to the n. So when we want to find our derivative function, we keep our constant A in front. And then we just differentiate our X to the n as we did before. So what happens is we bring down our n in front. That's why we get our A times n here. And then we minus one from the index up here. In our third formula here, if f of X is A to the X. So our function here is X to the power of one. So it's just X. We have our coefficient A. If we want to find the derivative f dash of X of this function, it's just A. It's just the constant in front of our X. And our special case here, when our f of X is a constant value A. So that's any constant value. The derivative of a constant function like this is always zero. So in the next video, we're going to apply these to some worked examples.