5. DIFFERENTIATION FORMULA
Derivative of Exponential Function
The derivative of the exponential function for
any given base and any differentiable function of u.
( )
f(x)uwhere;
dx
du
e)e(
dx
d
f(x)uwhere;
dx
du
alna)a(
dx
d
uu
uu
==
=
==
:ebaseFor
:abasegivenanyFor
12. A. Find the derivative and simplify the result.
( ) 1x3x2
3xg.1 +−
=
( )
22
xlnx
exf.2 +
=
2e
e
y.3 x3
x4
+
=
( )
3
x22
2 3xlogxh.4 ⋅=
( )
2
x
5xG.1 =
2lnyxxeye.2 22yx
++=+
( ) ( )
2
X
1xxH.3 +=
1x2
e
y.4
1x2
+
=
+
( ) ( )x2x2
eelnxf.5 −
+=
B. Apply the appropriate formulas to obtain the
derivative of the given function and simplify.
EXERCISES:
13. Logarithmic Differentiation
Oftentimes, the derivatives of algebraic functions
which appear complicated in form (involving
products, quotients and powers) can be found
quickly by taking the natural logarithms of both
sides and applying the properties of logarithms
before differentiation. This method is called
logarithmic differentiation.
14. 1. Take the natural logarithm of both sides and
apply the properties of logarithms.
2. Differentiate both sides and reduce the right
side to a single fraction.
3. Solve for y’ by multiplying the right side by y.
4. Substitute and simplify the result.
Steps in applying logarithmic differentiation.
Logarithmic differentiation is also applicable
whenever
the base and its power are both functions.
15. x
xyif
dx
dy
Find.1 =
xlnxyln
xlnyln x
=
=
Logarithmic differentiation is also applicable
whenever the base and its power are both functions.
(Variable to variable power.)
Example:
( ) ( )1xln1
x
1
x'y
y
1
+=
( ) x
xybutyxln1'y =→+=
( )( )x
xxln1'y +=∴
