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Derivative of an exponential function

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(i) The derivative of an exponential function with base e and variable x is the same exponential function multiplied by the derivative of x, which is 1. (ii) To take the derivative of an exponential function raised to a power, use the power rule and multiply the derivative of the exponential by the power. (iii) When adding exponential functions with the same base, the derivatives are added separately and the sums are multiplied by the common exponential term.

Derivative of an exponential function

Derivative of an Exponential Function.
Example-10:
(i)
x
ey =
Solution:
x
e
dx
dy
=
(ii)
x
ey 3
=
Solution:
( )3 3
3x xdy d
e e
dx dx
= =
(iii) ( )24
+= xey x
Solution:
( ) ( ) xx
e
dx
d
xx
dx
d
e
dx
dy 44
22 +++=
( ) ( ) ( )x
dx
d
exe xx
421 44
++=
( ) ( )42 44 xx
exe ++=
( )? ?4214
++= xe x
( )8414
++= xe x
( )944
+= xe x

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Derivative of an exponential function

2. Derivative of an Exponential Function. Example-10: (i) x ey = Solution: x e dx dy = (ii) x ey 3 = Solution: ( )3 3 3x xdy d e e dx dx = = (iii) ( )24 += xey x Solution: ( ) ( ) xx e dx d xx dx d e dx dy 44 22 +++= ( ) ( ) ( )x dx d exe xx 421 44 ++= ( ) ( )42 44 xx exe ++= ( )? ?4214 ++= xe x ( )8414 ++= xe x ( )944 += xe x