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Calculus volume 1
Derivatives
Derivatives of exponential and
Problem-solving strategy: using logarithmic differentiation
To differentiate
y
=
h
(
x
) using logarithmic differentiation, take the natural logarithm of both sides of the equation to obtain
ln
y
=
ln
(
h
(
x
)
)
.
Use properties of logarithms to expand
ln
(
h
(
x
)
) as much as possible.
Differentiate both sides of the equation. On the left we will have
1
y
d
y
d
x
.
Multiply both sides of the equation by
y to solve for
d
y
d
x
.
Replace
y by
h
(
x
)
.
Using logarithmic differentiation
Find the derivative of
y
=
(
2
x
4
+
1
)
tan
x
.
Use logarithmic differentiation to find this derivative.
ln
y
=
ln
(
2
x
4
+
1
)
tan
x
Step 1. Take the natural logarithm of both sides.
ln
y
=
tan
x
ln
(
2
x
4
+
1
)
Step 2. Expand using properties of logarithms.
1
y
d
y
d
x
=
sec
2
x
ln
(
2
x
4
+
1
)
+
8
x
3
2
x
4
+
1
·
tan
x
Step 3. Differentiate both sides. Use the
product rule on the right.
d
y
d
x
=
y
·
(
sec
2
x
ln
(
2
x
4
+
1
)
+
8
x
3
2
x
4
+
1
·
tan
x
)
Step 4. Multiply by
y
on both sides.
d
y
d
x
=
(
2
x
4
+
1
)
tan
x
(
sec
2
x
ln
(
2
x
4
+
1
)
+
8
x
3
2
x
4
+
1
·
tan
x
)
Step 5. Substitute
y
=
(
2
x
4
+
1
)
tan
x
.
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Using logarithmic differentiation
Find the derivative of
y
=
x
2
x
+
1
e
x
sin
3
x
.
This problem really makes use of the properties of logarithms and the differentiation rules given in this chapter.
ln
y
=
ln
x
2
x
+
1
e
x
sin
3
x
Step 1. Take the natural logarithm of both sides.
ln
y
=
ln
x
+
1
2
ln
(
2
x
+
1
)
−
x
ln
e
−
3
ln
sin
x
Step 2. Expand using properties of logarithms.
1
y
d
y
d
x
=
1
x
+
1
2
x
+
1
−
1
−
3
cos
x
sin
x
Step 3. Differentiate both sides.
d
y
d
x
=
y
(
1
x
+
1
2
x
+
1
−
1
−
3
cot
x
)
Step 4. Multiply by
y
on both sides.
d
y
d
x
=
x
2
x
+
1
e
x
sin
3
x
(
1
x
+
1
2
x
+
1
−
1
−
3
cot
x
)
Step 5. Substitute
y
=
x
2
x
+
1
e
x
sin
3
x
.
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Extending the power rule
Find the derivative of
y
=
x
r where
r is an arbitrary real number.
The process is the same as in
[link] , though with fewer complications.
ln
y
=
ln
x
r
Step 1. Take the natural logarithm of both sides.
ln
y
=
r
ln
x
Step 2. Expand using properties of logarithms.
1
y
d
y
d
x
=
r
1
x
Step 3. Differentiate both sides.
d
y
d
x
=
y
r
x
Step 4. Multiply by
y
on both sides.
d
y
d
x
=
x
r
r
x
Step 5. Substitute
y
=
x
r
.
d
y
d
x
=
r
x
r
−
1
Simplify.
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Key concepts
On the basis of the assumption that the exponential function
y
=
b
x
,
b
>
0 is continuous everywhere and differentiable at 0, this function is differentiable everywhere and there is a formula for its derivative.
We can use a formula to find the derivative of
y
=
ln
x
, and the relationship
log
b
x
=
ln
x
ln
b allows us to extend our differentiation formulas to include logarithms with arbitrary bases.
Logarithmic differentiation allows us to differentiate functions of the form
y
=
g
(
x
)
f
(
x
) or very complex functions by taking the natural logarithm of both sides and exploiting the properties of logarithms before differentiating.
Key equations
Derivative of the natural exponential function
d
d
x
(
e
g
(
x
)
)
=
e
g
(
x
)
g
′
(
x
)
Derivative of the natural logarithmic function
d
d
x
(
ln
g
(
x
)
)
=
1
g
(
x
)
g
′
(
x
)
Derivative of the general exponential function
d
d
x
(
b
g
(
x
)
)
=
b
g
(
x
)
g
′
(
x
)
ln
b
Derivative of the general logarithmic function
d
d
x
(
log
b
g
(
x
)
)
=
g
′
(
x
)
g
(
x
)
ln
b
For the following exercises, find
f
′
(
x
) for each function.
Questions & Answers
find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3
derivative of logarithms function
how to solve this question
sidra
ex 2.1 question no 11
khansa
anyone can help me
khansa
ex 2.1 question no. 11
khansa
Find the derivative of g(x)=−3.
any genius online ? I need help!!
need to learn polynomial
Zakariya
plz help me in question
Abish
evaluate the following computation (x³-8/x-2)
teach me how to solve the first law of calculus.
teach me also how to solve the first law of calculus
Bilson
only god knows😂
abdulkadir
f(x) = x-2
g(x) = 3x + 5
fog(x)?
f(x)/g(x)
fog(x)= f(g(x))
= x-2
= 3x+5-2
= 3x+3
f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
find the bounded area of the parabola y^2=4x and y=16x
what is absolute value means?
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |.
The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
y=(x^2 + 3x).(eipix)
Claudia
Source:
OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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