4. Find the derivative of the following functions using first principle.
(a) f(x) = e2x
(b) f(x) = x3 - 2
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Find the derivative using first principles? :
y
=
e
2
x
Calculus Limit Definition of Derivative
1 Answer
Steve M
Jan 5, 2018
d
y
d
x
=
2
e
2
x
Explanation:
Using the limit definition of the derivative:
y
=
f
(
x
)
⇒
d
y
d
x
=
lim
h
→
0
f
(
x
+
h
)
−
f
(
x
)
h
So if
y
=
e
2
x
; then:
d
y
d
x
=
lim
h
→
0
e
2
(
x
+
h
)
−
e
2
(
x
)
h
=
lim
h
→
0
e
2
x
+
2
h
−
e
2
x
h
=
lim
h
→
0
e
2
x
e
2
h
−
e
2
x
h
=
lim
h
→
0
e
2
x
(
e
2
h
−
1
)
h
=
e
2
x
lim
h
→
0
e
2
h
−
1
h
=
e
2
x
lim
h
→
0
2
(
e
2
h
−
1
)
2
h
=
2
e
2
x
lim
h
→
0
e
2
h
−
1
2
h
Then if we perform, a substitution,
α
=
2
h
then clearly:
h
→
0
⇒
α
→
0
So we can write the derivative as:
d
y
d
x
=
2
e
2
x
lim
α
→
0
e
α
−
1
α
Now
lim
α
→
0
e
α
−
1
α
=
1
is a standard calculus limit whose limit is unity, and so we find that:
d
y
d
x
=
2
e
2
x
Answer link
Related topic
Limit Definition of Derivative
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