Differentiating exponentials sheet - University of Exeter

PHY1106: Waves and Oscillators: Dr. Pete Vukusic

Differentiating exponentials

The exponential function ex is perhaps the easiest function to differentiate: it is the only function whose derivative is the same as the function itself.

d (ex ) = ex dx

d dx

(eu )

=

ex

du dx

Strictly speaking, this is the correct general case

Slightly more complicated is e2x or e3x and so on. In these cases the number in front of the x "comes down" to the front in the derivative, (this is because we actually "bring down" the differential of this exponent)

d dx

(e2x

)

=

2e2x

and

d dx

(e3x

)

=

3e3x

For example, if the "exponent" is now a slightly more complicated function of x, see how we "bring down" the differential of this function; e.g. for exp (x2) etc.

d (ex2 ) = 2x.ex2 dx

and

d (ex3 ) = 3x2.ex3 dx

If the function is in terms of time (t) instead of x, (as with some of the functions in our lecture course, e.g. x = exp(t) ), then the differential works in the same way;

d (et ) = et dt

Or for the complex displacement of our SHM system; i.e. x = A.exp j(t + )

x& = d (x) = d (Aej(t+) ) = Ajej(t+) dt dt

Differential of raised exponent

Practice Problems:

Complete the following:

d dx

(e2 x

2

)

=

d dx

(ex 2

+2x

)

=

d dt

(Ae j(t +)

)

=

d dt

(Aje j(t+)

)

=

d dt

(-

je j(2t +) )

=

d dt

Aje- j(t +)

=

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