Exponential and Logarithmic Functions & Inverse Derivative

If y=lnx, then y'=

1/x

If y=lnu, then y'=

u'/u

If y=eˣ, then y'=

eˣ

If y=eᵘ, then y'=

u'×eᵘ

If y=loga(x), then y'=

1/(xlna)

If y=loga(u), then y'=

u'/(ulna)

If y=aˣ, then y'=

aˣ(lna)

If y=aᵘ, then y'=

aᵘ×u'×lna

ln1

0

e⁰

1

lne

1

lneˣ

x

Inverse Derivative Steps

1 Find f'(x)
2 Set f(x) to a & solve for x, x=b
3 Find f'(b)
4 The (f^-1)'(a)=1/f'(b)

Let f be a function that is differentiable on an interval I. If f has an inverse function then the following statements are true:

1. If f is continuous on its domain, then f⁻¹ is continuous on its domain
2. If f is increasing on its domain, then f⁻¹ is increasing on its domain.
3. If f is decreasing on its domain, then f⁻¹ is decreasing on its domain.
4. If f is differentiable at c and f'(c)≠0, then f⁻¹ is differentiable at f(c).

Steps to show that the slopes of the graphs f(x) and f^-1(x) at (a,b) and (b,a) are recirpocals

1 Find the derivatives of both
2 Plug in the x-component of each of the corresponding points into the derivatives
3 Verify that they are reciprocals

(arcsinu)' =

u'/√1-u²

(arccosu)'=

-u'/√1-u²

(arctanu)'=

u'/(1+u²)

(arccotu)'=

-u'/(1+u²)

(arcsecu)'=

u'/|u|√u²-1

(arccscu)'=

-u'/|u|√u²-1

arcsin(x) can be simplified to...

sinθ = x

arccos(x) can be simplified to...

cosθ = x

arctan(x) can be simplified to...

tanθ = x

arccsc(x) can be simplified to...

sinθ = 1/x

arcsec(x) can be simplified to...

cosθ = 1/x

arccot(x) can be simplified to...

tanθ = 1/x

Properties of Natural Exponential Functions

1. Range: (0,∞)
2. Domain: (-∞,∞)
3. The function is continuous, increasing and one-to-one
4.The graph is concave up
5. lim eˣ as x→-∞ = 0 and lim eˣ as x→∞ = ∞