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 =

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

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

=

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

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