Differentiation Flashcards
What does lnx differentiate to?
1/x
When a function is increasing, what does that mean?
dy/dx>0, gradient is positive
When a function is decreasing, what does that mean?
dy/dx<0, gradient is negative
How to differentiate natural log functions
e^x = e^x
e^kx = ke^kx (power never changes basically)
cosx differentiates to
-sinx
-sinx differentiates to
-cosx
-cosx differentiates to
sinx
sinx differentiates to
cosx
lnkx+c differentiates to
k/kx+c
a^x differentiates to
a^xlna, by taking natural logs and implicit differentiation
a^kx differentiates to
ka^kxlna
sin(kx) differentiates to
kcos(kx)
cos(kx) differentiates to
-ksin(kx)
product rule
u(dv/dx) + v(du/dx)
differentiate arcsinx
1/(√1-x^2)
differentiate arccosx
-(1/(√1-x^2))
differentiate arctanx
1/(1+x^2)
How do you differentiate arc trig functions?
use the rule dy/dx is reciprocal of dx/dy
if y = arctrigx, then x = trigy
How to differentiate parametric functions?
Use (dy/dt)/(dx/dt) = dy/dx. differentiate each separately with respect to t.
How to test whether a point is a point of inflection?
See if second differential = 0, then see if sign changes on second differential either side of test point.
How to implicitly differentiate xy?
Use product rule.
How do you differentiate sin^2x?
Imagine it as (sinx)^2 and use chain rule.
If a function is concave, what does that mean?
Second derivative is positive
If a function is convex, what does that mean?
Second derivative is negative
What does ln2x differentiate to?
1/x
