Flashcards
Differentiation Flash Cards
Cards in this deck (17)
- 1What is the basic rule of differentiation?If f(x) = x^n then f'(x) = nx^(n-1)
- 2First Step in the Chain Rulelet the u=the bracketed component
e.g. y = (x^4 + 3x-7)^5
Let u = x^4 + 3x-7
y=u^5 - 3Second Step Chain Ruledifferentiate both expressions, u= and y=
e.g. du/dx = 4x^3 +3
dy/du = 5u^4 - 4Third Step Chain Ruledy/dx = dy/du x du/dx
e.g. dy/dx = 5u^4 x (4x^3 + 3)
Then expand...
dy/dx = 5(x^4 +3x -7)^4 x (4x^3+3) - 5How do you implement the Product Rule?[z x w]' = [z' x w] + [z x w']
- 6How do you implement the Quotient Rule?(z/w) = [(z x w') - (z' x w)] / [w^2]
- 7how do you differentiate f(x) = e^x?f (x) = ae^(bx)
f'(x) = (bx')ae^(bx)
Change nothing except multiply by the differentiated power - 8How do you differentiate ln?f(x) = ln[g(x)]
f'(x) = g'(x) x 1/g(x) = g'(x)/g(x)
e.g. f(x) = ln(2x^3 + 5)
f'(x) = 6x^2 x 1/(2x^3 + 5) = (6x^2)/(2x^3 + 5) - 9How do you differentiate any number to the power of x?If f(x) = a^x
then f'(x) = lna x a^x - 10What are the differentials of trigonometric functions?sinx = cosx
cosx = -sinx
tanx = sec^2x
cosecx = -cosecxcotx
secx = secxtanx
cotx = -cosec^2x - 11How do you differentiate a trigonometric function with multiple angles?e.g. If f(x) = sin5x
then f'(x) = cos5x
but
If f(x) = sec5x
then f'(x) = 5sec5xtan5x - 12How do you differentiate trigonometric functions with powers?If f(x) = sin^2x that equals sinxsinx
Use product rule
this means f'(x) = 2sinxcosx - 13How to implement basic Parametric Differentiation?dy/dx = (dy/dt) x (dt/dy)
- 14First step Parametric Differentiation?differentiate x and y
e.g. If x=t^2
then dx/dt = 2t
If y= t-1/t
then dy/dt = 1 + 1/t^2 = (t^2 + 1) / t^2 - 15Second step Parametric Differentiation?flip dx/dt to become dt/dx
e.g. dx/dt = 2t
dt/dx = 1 / 2t - 16Third step Parametric Differentiation?dy/dt x dt/dx = dy/dx
e.g. [(t^2 + 1) / (t^2)] x 1/2t = (t^2 +1) / 2t^3 - 17What can 1 be substituted for?1= a/a
e.g. 1= 2t/2t
This can be useful to fully simplify an equation like 1+ 1/t^2
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- Description
- Flash Cards for the most important differentiation equations and rules
- Created by
- Oskar Simmers
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