Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Derivative of a function definition. The derivative of the function F of X with respect to the variable X is the function F prime whose value at X is F prime of X equal. The limit is H goes to 0 of F of X + H -, F of X all over H. The alternative form for the derivative is F prime of X equals the limit as Z approaches X of F of Z -, F of X / Z -, X. That really, really really is just two points Z&F of Z and XF of X. The process of calculating a derivative is called differentiation. There are many different notations for differentiation. Notations for derivative of function Y equal F of XF prime F prime of XY prime the derivative of Y in terms of the derivative of X. The derivative of the function Y in terms of X. The derivative of the function with respect to X. The derivative in terms of X of the function F of XD of F of X and DXF of X. The reason there are so many notations is that if you study enough different levels of calculus, you will Start learning how to take derivatives in terms of other variables besides X. So DDX and D are called differential operators. To indicate the value of a derivative at a specified number X equal a, we write F prime of a equal dy DX at X equal a or DFDX at X equal a or the derivative in terms of X of F of X when X approaches or when X = a. What can we learn from the graph of Y equal F prime where the rate of change of F is positive, negative, or zero? The rate of change being our slope, the rough size of the growth rate at any X and its size in relation to the size of F of X and where the rate of change itself is increasing or decreasing. So if we look at an example, we want to find the derivative of a line, so the derivative of MX plus B. Using our definition, we're going to stick an X + H for the X. So F of X + H is the quantity M times the quantity X + H + b minus our F of X equation MX plus B all divided by H If we distribute, we see that the MX and the B's are going to cancel and then in the next step our HS are going to cancel. So the limit as H goes to 0 of M there is no H so that's just going to be the constant M. So the derivative of any line is MA function has a derivative at a point if and only if it has a left hand and right hand derivative there, and these one sided derivatives are equal. So the derivative of MX plus B equal M. We just showed that the derivative in terms of the square root of X is going to be 1 / 2 square roots of X. We could prove that. Similarly, when a function does not have a derivative at a point when there's a jump discontinuity, or a corner, or a cusp of vertical tangent and a removable discontinuity, we have a theorem that says differentiability implies continuity. If F has a derivative at X = C, then F is continuous at X = C Caution, a function need not have a derivative at a point where it is continuous. That's important. Frequently people confuse this. It says if it has a derivative, it's continuous. It does not say if it's continuous, it has a derivative. So the proof if H doesn't equal 0, then we know that F of C + H is really just equaling F of C + F of C + H -, F of C. Basically all I did was I added an F of C and I subtracted an F of C because I'm trying to get it into one of the formulas that we're familiar with. So I haven't done anything to that. FSC plus HI added an FSC and I subtracted an FSC. Now we're going to take the FSC plus H minus FSC part and divide by H But if I divide an equation by H, you're going to hopefully tell me we have to multiply the equation by H. So now if we take the limit as H goes to zero, we have the limit as H approaches zero of the original FSC plus H, which is really just equal to the limit of F of C as H approaches 0 plus the limit as F of C + H -, F of C all over H as H approaches zero. Well, the limit as H approaches zero of F of C + H is equal to if there is no H. Here, the limit of a constant is just whatever it was, so in this case it's F of C plus. This by definition is the derivative of the F primacy. And then we had this limit as H goes to 0 out here of H By sticking 0 for H, that's just 00 times anything, so that F prime C * 0 is going to go away. So we've just proven that the limit as H goes to 0 of F of C + H really does equal F of C So if F has a derivative at X = C, then F is continuous at X = C. Thank you and have a wonderful day.