Core 3 - Differentiation Flashcards
What is the differential of sin X
Cos X
What is the differential of tan X and tan(4x)
Sec^2 X.
d/dx (tan(4x) = 4sec^2(4x)
What is he differential of cos X
-sin X
What is the chain rule used to differentiate and what is the general formula for dy/dx?
Give 2 examples of equations differentiated by chain rule.
Function of a function e.g. y=e^x, y=(2x+3)^2
dy/dy = dy/du * du/dx
How would you differentiate y=(7x^2+5x-8)^5
a) by chain rule
b) by inspection
a) u=7x^2+5x-8
therefore y=(u)5
du/dx = 14x+5
dy/du = 5u4
dy/dx = (70x+25)(7x^2+5x-8)4
b) take power to the front, reduce power by 1
Multiply my differential of brackets
Leave brackets as they are.
What is the product rule used to differentiate and what is the general formula for dy/dx?
Product rule used to differentiate y=uv,
dy/dx = v(du/dx)+u(dv/dx)
What is the general formula for the differentiation, by inspection of y=e^(f(x))?
dy/dx = f’(x) *e^(f(x))
What is the general formula for differentiating, by inspection, y=ln(x)? What is recognising this form useful for?
dy/dx =f’(x)/f(x)
This is useful when recognising integrations, if the original equation is in this form, the integral is a ln
What do you use the quotient rule to differentiate? What is the formula?
One function divided by another e.g. (2x+3)/e^x.
dy/dx = (v(du/dx)-u(dv/dx))/v^2
What do you use the chain rule to differentiate? What is the formula?
Function of a function, e.g. (2x+3)^2 –> 4(2x+3)
dy/dx = dy/du * du/dx
What is the differential of 2sin(3x)
6cos(3x)
What is the differential of 18cos(0.5x)
-9sin(0.5x)
What is the differential of -8sin(2x)
-16cos(2x)
What is the differential of xsin(3x)
(X*3cos(3x)+(sin(3x))
