Exponential and logs

1. IB MSL Page 1
x xdy
y ke ke
dx
If = then =
The Derivative of ex
A special property of the exponential function ex
is that
Example 1 Find the derivative of y = 4ex
– x3
Example 2 Differentiate y = e5x
with respect to x
x xdy
y e e
dx
If = then =

2. IB MSL Page 2
More generally
If you are a visual leaner the above looks like this:
Using to representf(x) and to represent f ’(x):
Example 3
= =
dy
y e e
dx

f f
f( ) ( )
If = then = '( )x xdy
y e x e
dx
5 4
( ) =xdy
e
dx


3. IB MSL Page 3
The Derivative of ln x
Remember, ln x is the inverseof ex
So, if y = ln x rewrite this in terms of x as the subject
Differentiating with respectto y gives
X =
ydx
e
dy
=
1 1
= = ydx
dy
dy
dx e
dy
dx x
1
=

4. IB MSL Page 4
f
f
f
'( )
If = ln ( ) then =
( )
dy x
y x
dx x
Example 3 Differentiate y = ln 3xwith respect to x
We can use the chain rule to extend to functions of the more general form
y = ln f(x). Use the chain rule to find the derivativeof y = ln f(x)?
1
If = ln then =
dy
y kx
dx x

5. IB MSL Page 5
ln(7 4) =
d
x
dx

4
2
lnx
x
Practice questions
Give the coordinates of any stationary points on the curve y = x2
e2x
Find the equation of the tangent to the curvey = at the point (1, 0).
3
ln(3 + 8) =
d
x
dx