Question

Can someone define what differentiation means? How something is differentiable. Maybe include how it relates to graphs.

I learned this in class last month but have been going to Calculus never really understanding the term. :(

Answer 1

the derivative of a straight line , is equal to the gradient of that line

Answer 2

So your derivative is basically an equation that gives you the slope at any point x on the graph. To be differentiable, the function HAS to be continuous, because how can you have a slope if your function doesn't exist there? So for example, you have the equation \[x ^{2}+3x+1\]. Now the derivative would be \[2x+3\]. This makes it so that you can find the slope at any point of x that you plug in.

Answer 3

So a graph has to be continuous because you cannot find the slope of the tangent line of a broken graph, correct?

Answer 4

Well think about it. If the graph doesn't exist there, there's no point for you to find a slope for, yeah?

Answer 5

Yeah.
Also, the reason why some graphs are not differentiable even though it is continuous is because of cusps. This is because cusps mean that there are many possible tangent lines at the pointy part of the cusp right? (this question is just to clarify)

Answer 6

Well because just like limits (assuming you've already done limits), if the limit from either side doesn't agree, the limit doesn't agree, right? Same thing with derivatives. When you look at the derivative on the left and right side of the point, they aren't the same, so you can't have a derivative for that point.

Answer 7

Hmm, never thought of it in terms of limits like that. Makes sense. Thanks! :D

Answer 8

Mhmmm :)