Lecture 3- Differentiation; Derivatives and Differentials; Convexity and Concavity Flashcards
What is the difference between derivative and differential?
dy is the differential of y due to a change in x … dy = dy/dx dx- JUST MULTIPLY THE DX OVER ON THE OTHER SIDE
dy/dx is the derivative of y with respect to x
What must you remember about the quotient rule?
V du/dx first and all over V^2
What is the chain rule?
dy/du * du/dx
What must you remember about the product rule?
You add the 2 parts of the equation
What is y=e^x also equal to?
lny=x as lne^x becomes x as lne cancels out
How would you describe a concave curve?
N shaped curve- also known as concave downwards
The slope decreases as x increases
It is wrong to describe a concave function as “one that gets flatter”
How would you describe a convex curve?
U shaped curve- also known as concave upwards
The slope increases as x increases
It is wrong to describe a convex function as “one that gets steeper”
How do you prove if a curve is concave or convex?
Find the 2nd derivative
If the 2nd derivative is greater than 0 then the curve has a minimum point in which case it is likely to be convex
If the 2nd derivative is less than 0 then the curve has a maximum point in which case it is likely to be concave
How would you generally differentiate a^x?
The derivative would be a^xlna
What would be the 1st order derivative of 3^x?
3^xln3
What should you do if you come across a function which you need to differentiate but it looks quite weird or odd?
Try and use the chain rule
