Differentiation Flashcards
Chain rule
Y= (f(x))^n
Dy/dx = n f’(x) (f(x))^n-1
Ditty for the product rule
1st x differential of the 2nd + 2nd x differential of the 1st
Ditty for the quotient rule
(Bottom x differential of the top - top x differential of the bottom) all divided by the bottom^2
E^x
E^x
E^f(x)
F’(x) e ^f(x)
Lnx
1/x
Ln f(x)
F’(x)/ f(x)
A^x
A^x lna
A^f(x)
F’(x) Lna af(x)
Sin x
Cos x
Cos x
-sin x
Sin f(x)
F’(x) cos f(x)
Cos f(x)
-f’(x) sin f(x)
Tan x
Sec^2 x
Tan f(x)
F’(x) sec^2 f(x)
Cot x
-cosec^2 x
Cot f(x)
-f’(x) cosec ^2 f(x)
Sec x
Sec x tan x
Sec f(x)
F’(x) sec f(x) tan f(x)
Cosec x
-cosec x cot x
Cosec f(x)
-f’(x) cosec f(x)cot f(x)
When do you use the chain rule
When the function you are differentiating is to a power
When do you use the product rule
When 2 functions are multiplying by each other
When do you use the quotient rule
If one function is dividing another function
Small angle approximation for sin theta (radians)
Theta
Small angle approximation for tan theta (radians)
Theta
Small angle approximation for cos theta (radians)
1- theta^2/2
Differentiating sin x using first principles
Substitute the f(x+h) with sin (x+h)
This is then the sin angle formula so will be sinxcosh + sinhcosx then -sinx on the end all over h
Factorise them so you have your sin x (cos h-1) on one side and the cosx sinh on the other
Cosx x 1 so f’(x)= cos x
Differentiating cos x using first principles
Set up using formula
Then get the cos angle formula -cosx
Factories so the cosx (cosh - 1) then - sinx sin h
So you get f’(x)= -sinx
If the trig function begins with co what will it differentiate to
A negative
How do you differentiate trig with powers
Use the chain rule
Describe how to differentiate y=7^x
Write it as e^ln 7^x Bring the x down and in front so e^xln7 Differentiate so ln7 e^x ^ln7 Put the x back so ln7 e ^ln7^x E and th ln cancel out so you’re left with ln7 . 7^x
How do you prove how to differentiate tan x
Tan x = sinx / cosx
Then use the quotient rule
How do you prove how to differentiate cot x
Cotx = cosx/sinx
Then use the quotient rule
How do you prove how to differentiate secx
Y=1/cos x
Then use the chain rule to the power of -1
How do you prove how to differentiate cosec x
Y= 1/sin x
Then use the chain rule to the power of -1
What do you do if the function is given in terms of y
Differentiate with respect to y
Dx/dy
What do you do if the function is given in terms of y but is asking for dy/do
Find dx/ dy
Then reciprocal
Differentiate y= arcsin x
1/ square root of (1-x^2)
Differentiate y= arccos x
-1/ square root of (1-x^2)
Differentiate y= arctan x
1/ 1+x^2
What is an implicit equation
Where x and y are muddled together and can’t be separated
How do you differentiate an implicit equation
Differentiate the y with respect to x and put dy/dx on the end
Often uses the product rule as the x and y are stuck together
How do you find the turning point of an implicit equation
Find dy/dx
Make equal to zero
Always keep the dy/dx on the left, move anything that has this asa factor onto the left, factorise then make it the subject
When d2y/dx^2 > 0
Minimum turning point
When d2y/dx^2 <0
Maximum turning point
When dy/dx >0
Increasing function
When dy/dx <0
Decreasing function
Convex
Minimum point
When d2y/dx^2 > 0
Concave
Maximum point
When d2y/dx^2 <0
Stationary point of inflection
Dy/dx = 0
And d2y/dx^2 changes sign
Non stationary point of inflection
Convex section where d2y/dx^2 > 0
Point of inflection where d2y/dx^2 changes sign
Concave section where d2y/dx^2 < 0
What is a point of inflection
Where f’’(x) changes sign
D2y/dx^2 will equal zero but you need to show that it changes sign before and after this point
