6.3. Differentiation and Integration of Logarithmic and Exponential Functions http://www.ck12.org
d
dx[bu] =bu·lnb·du
dx.And ifb=e,
d
dx[eu] =eu·du
dx.Derivatives of Exponential Functions
d
dx[bu] =bu·lnb·du
dx=u′bulnb
d
dx[eu] =eu·du
dx=u′euExample 9:
Find the derivative ofy= 2 x^2.
Solution:
Applying the rule for differentiating an exponential function,
y′= ( 2 x) 2 x^2 ln 2
= 2 x^2 +^1 ·x·ln 2.Example 10:
Find the derivative ofy=ex^2.
Solution:
Since
d
dx[eu] =u′eu,
y′= 2 xex^2.Example 11:
Findf′(x)if
f(x) = √^1 πσe−αk(x−x^0 )^2.whereσ,α,x 0 ,andkare constants andσ 6 = 0.
Solution:
We apply the exponential derivative and the Chain Rule: