2023-2024
F.4 Mathematics Module II
Ch.5 Differentiation
Lesson WS
Name:____________________( )
5.1 Derivatives
Tangent to the Curve
Definition: A tangent line is a line that touches a curve at a single point and does not cross through it.
The point where the curve and the tangent meet is called the point of tangency.
Slope of a Curve
When we say the slope of a curve, it means the slope of tangent to the curve at a point.
1. Start with what we can calculate,
namely the slope of secant PQ.
i.e. slope of PQ
=__________________________
2. Investigate the limiting value of the
secant slope as Q approaches P along
the curve.
3. If the curve is continuous, the limit exists,
take it to be the slope of the curve at P
i.e. slope of tangent to the curve at P
=______________________________________
Slope of Tangent to the Curve
Definition:
The slope of tangent to the curve y = f (x) at the point is given by
The tangent line to the curve at P is the line through P with this slope.
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Example 5.11
Find the slope of the tangent to the curve at
(a) x = 1, (b) x = 2.
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Quick Practice 5.11
Find the slope of the tangent to the curve at x = 1.
Definition
The derivative of a function y = f(x) with respect to x, denoted by , is defined as
, provided that the limit exists.
Remarks:
The process of finding the derivative of a function is called differentiation. Find the derivative of a
function using is called finding the derivative of a function from first
principles.
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Remarks:
1. If the limit of the function y = f(x) does not exist at a particular value of x, then there is no
derivative of y = f(x) at this value of x.
2. We may denote the derivative of y = f(x) with respect to x by , , or .
3. The derivative of y = f(x) at x = a is denoted by or , which is the slope of y = f(x) at
x = a.
4. If the derivative of y = f(x) at x = a can be found, then y = f(x) is said to be differentiable at x = a.
5. is a symbol instead of a fraction.
Example 5.12
Find for the following functions from first principles.
(a) (b)
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Quick Practice 5.12A
Find for the following functions from first principles.
(a) (b)
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Quick Practice 5.12B
Find for the following functions from first principles.
(a) (b)
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Quick Practice 5.12C
Find for the following functions from first principles.
(a) (b)
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Example 5.13
(a) Find the derivative of with respect to x from first principles.
(b) Hence, find the slope of the tangent to the curve at x = 1.
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Quick Practice 5.13
(a) Find the derivative of with respect to x from first principles.
(b) Hence, find the slope of the tangent to the curve at x = 11.
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5.2 Basic Rules of Differentiation
Differentiation of Constant Functions and Power Functions
Theorem 5.1
If f(x) = k, where k is a constant, then
Theorem 5.2
If , where n is a non-zero real number, then .
i.e.
Prove Theorem 5.1 and 5.2 from First Principles
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Differentiation of a Constant times a function
Theorem 5.3
If f(x) is a differentiable function and k is a constant, then
Differentiation of Sum and Difference of Functions
Theorem 5.4 (Addition Rule)
If f(x) and g(x) are differentiable functions of x, then
(a)
(b)
Prove Theorem 5.3 and 5.4 from First Principles
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Example 5.21
Find the derivatives of the following functions with respect to x.
(a) (b)
Quick Practice 5.21
Find the derivatives of the following functions with respect to x.
(a) (b)
Example 5.22
Find the derivatives of the following functions with respect to x.
(a) (b)
Quick Practice 5.22
Find the derivatives of the following functions with respect to x.
(a) (b)
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Example 5.23
Find the derivatives of the following functions with respect to x.
(a) (b)
Quick Practice 5.23A
Find the derivatives of the following functions with respect to x.
(a) (b)
Quick Practice 5.23B
Find the derivatives of the following functions with respect to x.
(a) (b)
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Differentiation of the Product of Two Functions
Theorem 5.5 (Product Rule)
If u = f(x) and v = g(x) are differentiable functions of x, then
Mnemoics : _________________________________
Prove Theorem 5.5 from First Principles
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Example 5.24
Differentiate with respect to x.
Quick Practice 5.24A
Differentiate with respect to x.
Quick Practice 5.24B
If , find .
Example 5.25
Let .
(a) Find .
(b) Find the value of .
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Quick Practice 5.25A
If , find .
Quick Practice 5.25B
Let .
(a) Find .
(b) Find the value of .
Quick Practice 5.25C
Find .
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Differentiation of the Quotient of Two Functions
Theorem 5.6 (Quotient Rule)
If u = f(x) and v = g(x) are differentiable functions of x, and g(x) ≠ 0, then
Mnemoics : _________________________________
Prove Theorem 5.6 from First Principles
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Example 5.26
Find for each of the following functions.
(a) (b)
Quick Practice 5.26
Find for each of the following functions.
(a) (b)
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Example 5.27
Given that , find the value of .
Quick Practice 5.27
Given that , find the value of .
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5.3 Differentiation of Composite Functions
Theorem 5.7 (Chain Rule)
Let y = f(u) and u = g(x). If y is a differentiable function of u and u is a differentiable functions of x, then
Mnemoics : _________________________________
Textbook use the method of “Let y = f(u) and u = g(x)”.
Example 5.31
Find for each of the following functions.
(a) (b)
Quick Practice 5.31
Find for each of the following functions.
(a) (b)
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Example 5.32
Differentiate the function with respect to x.
Quick Practice 5.32
Differentiate the function with respect to x.
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Example 5.33
Find the derivatives of the following functions with respect to x.
(a) (b)
Quick Practice 5.33A
Find .
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Quick Practice 5.33B
Differentiate with respect to x.
Quick Practice 5.33C
Find .
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Example 5.34
Let . Find the value of .
Quick Practice 5.34
Let . Find the value of .
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5.4 Differentiation of Trigonometric Functions
Theorem 5.8
(a) (b)
Where x is measured in radians.
Prove Theorem 5.8 from First Principles
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Theorem 5.9
, where x is measured in radians.
Proof of Theorem 5.9
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Example 5.41
Find for each of the following functions.
(a) (b)
Quick Practice 5.41
Find for each of the following functions.
(a) (b)
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Example 5.42
Find for each of the following functions.
(a) (b) (c)
Quick Practice 5.42
Find for each of the following functions.
(a) (b) (c)
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Example 5.43
Find the value of for each of the following.
(a) (b)
Quick Practice 5.43
Find the value of for each of the following.
(a) (b)
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Enrichment Knowledge
(a) Prove that , where x is measured in radians.
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Enrichment Knowledge
(b) Prove that , where x is measured in radians.
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Enrichment Knowledge
(c) Prove that , where x is measured in radians.
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5.5 Differentiation of Exponential Functions and Logarithmic Functions
Differentiation of Exponential Functions
Theorem 5.10
Proof:
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Differentiation of Logarithmic Functions
Theorem 5.11
, where x > 0
Proof:
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Example 5.51
Find for each of the following functions.
(a) (b) (c) (d)
Quick Practice 5.51A
Find for each of the following functions.
(a) (b) (c)
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Quick Practice 5.51B
Find the derivative of the function with respect to x.
Example 5.52
Find the derivative of the function with respect to x.
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Quick Practice 5.52A
Find the derivative of the function with respect to x.
Quick Practice 5.52B
Find the derivative of the function with respect to x.
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Example 5.53
Find the derivative of the following functions with respect to x.
(a) (b) (c)
Quick Practice 5.53A
Find the derivative of the following functions with respect to x.
(a) (b) (c)
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Quick Practice 5.53B
Find the derivative of the following functions with respect to x.
(a) (b)
Example 5.54
Differentiate the following functions with respect to x.
(a) (b)
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Quick Practice 5.54A
Differentiate the following functions with respect to x.
(a) (b)
Quick Practice 5.54B
Differentiate the following functions with respect to x.
(a) (b)
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5.6 Implicit Differentiation and Logarithmic Differentiation
Implicit Differentiation
Not every implicit function can be rewritten as an explicit function easily. In order to compute for
those implicit functions, we have to apply the technique of implicit differentiation by the following steps:
(1) Differentiate both sides of the given equation with respect to x, regarding y as a function of x.
(2) Express in terms of x and y.
Let’s Practice
Express the following derivatives in terms of x, y and .
(a) (b)
(c) (d)
(e) (f)
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Example 5.61
Find for each of the following functions.
(a) (b)
Quick Practice 5.61
Find for each of the following functions.
(a) (b)
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Example 5.62
Given that , find .
Quick Practice 5.62
Find for each of the following functions.
(a) (b)
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Example 5.63
It is given that B(0, m) is a point on the curve .
(a) Find the value of m.
(b) find the value of at B.
Quick Practice 5.63
It is given that P(k, 0) is a point on the curve .
(a) Find the possible value(s) of k.
(b) Find the value(s) of at P.
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Logarithmic Differentiation
Let y = f(x). if f(x) can be expressed as products, quotients or powers of algebraic functions, the method of
logarithmic differentiation is a useful technique to find . Logarithmic differentiation can be done as
follows:
(1) Take natural logarithms on both sides of y = f(x).
(2) Differentiate both sides of the equation with respect to x.
(3) Express in terms of x only.
Example 5.64
If ,
(a) find .
(b) where , find by logarithmic differentiation.
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Quick Practice 5.64
If where x > 5, find by logarithmic differentiation.
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Example 5.65
If ,
(a) find .
(b) where , find by logarithmic differentiation.
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Quick Practice 5.65
If where x > 0, find by logarithmic differentiation.
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Logarithmic Differentiation
Logarithmic differentiation is mainly used to find derivatives of functions in the form .
Example 5.66
If where x > 0, find .
Quick Practice 5.66
If , find .
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5.7 Second Derivatives
For a differentiable function y = f(x), is the first derivative of y with respect to x.
If is also differentiable, then the derivative of with respect to x is called the second derivative of
y with respect to x and it is denoted by , or .
Remark:
Example 5.71
Find the second derivative for each of the following functions with respect to x.
(a) (b)
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Quick Practice 5.71
Find the second derivative for each of the following functions with respect to x.
(a) (b)
Example 5.72
Let . Find the value of .
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Quick Practice 5.72
Let . Find the value of .
Example 5.73
Consider . Suppose m is a constant such that for all real values of
x. Find the value of m.
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Quick Practice 5.73
Let , where m is a constant. If for all real values of x, find the
value of m.
Example 5.74
If , show that .
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Quick Practice 5.74
If , show that .
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Summary (hand it in on GC)
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