Breakout Session 13: Derivatives of
inverse functions
A look back: In the previous (February 18, 2016) Breakout Session you practiced differentiating implicit equations and geometrically interpreting dy/dx.
Overview: In today’s (February 23, 2016) Breakout Session you’ll practice
differentiating inverse (trigonometric) functions and the connection between the
derivative of f −1 with the derivative of f .
A look ahead: In the next (February 25, 2016) Breakout Session you be introduced to related rates and take the computation derivatives quiz.
Learning Outcomes
The following outcomes are not an exhaustive list of the skills you will need to
develop and integrate for demonstration on quizzes and exams. This list is meant
to be a starting point for conversation (with your Lecturer, Breakout Session
Instructor, and fellow learners) for organizing your knowledge and monitoring
the development of your skills.
• Take derivatives of logarithms and exponents of all bases.
• Take derivatives of functions raised to functions.
• Apply the generalized power rule.
• Recognize the difference between a variable as the base and a variable as
the exponent.
• Identify situations where logs can be used to help find derivatives.
• Use logarithmic differentiation to simplify taking derivatives.
• Find derivatives of inverse functions in general.
• Understand how the derivative of an inverse function relates to the original
derivative.
• Derive the derivatives of inverse trig functions.
• Recall the meaning and properties of inverse trig functions.
• Take derivatives which involve inverse trig functions.
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Breakout Session 13: Derivatives of inverse functions
Facts to know and eventually remember
(1) eln x = x for x > 0 and ln(ex ) = x for all x.
(2) y = ln x ⇐⇒ ey = x.
(3) For every x and b > 0 we have bx = ex ln b .
(4) ln(xy) = ln(x) + ln(y).
(5) ln(x/y) = ln(x) − ln(y).
(6) ln(xz ) = z ln(x).
(7) ln(e) = 1.
Problem 1 True or False:
(1) If f (x) = (x − 2)x , then f 0 (x) = x(x − 2)x−1 .
(2) If f (x) = (3x)x , then f 0 (x) = (3x)x ln(3x).
Problem 2 Find the derivatives of the following functions:
x
(a) f (x) = xe + 7x
4
(b) g(x) = (ln x + 9)sec(x
(c) h(x) =
)
(x2 − 7)5
cos7 (x2 − 5)
Problem 3 A table of values for f and f 0 is shown below. Suppose that f is a
one-to-one function and f −1 is its inverse.
x
f (x)
f 0 (x)
1
3
4
3
4
6
4
5
3
(I) Evaluate f −1 (f (x)) at x = 3.
(a) 1
(b) 3
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Breakout Session 13: Derivatives of inverse functions
(c) 6
(d) 4
(e) DNE
(f) None of the previous answers
(II) Evaluate
d
f (f (x)) at x = 3.
dx
(a) 6
(b) 25
(c) 5
(d) 15
(e) DNE
(f) None of the previous answers
(III) Evaluate
d
ln((f (x)) at x = 3.
dx
(a) 1/4
(b) 5
(c) 5/4
(d) 1/5
(e) DNE
(f) None of the previous answers
(IV) Evaluate f −1 (x) at x = 3.
(a) 4
(b) 1
(c) 1/3
(d) 5
(e) DNE
(f) None of the previous answers
(V) Evaluate
d −1
f (x) at x = 3.
dx
(a) 1
(b) 4
(c) 1/5
(d) 1/4
(e) 5
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Breakout Session 13: Derivatives of inverse functions
(f) None of the previous answers
Problem 4 Find the derivatives of the following functions:
√
(a) f (x) = sec−1 ( x).
(b) g(x) = ln(sin−1 (x)).
1
.
tan−1 (x2 + 4)
(c) h(x) =
Problem 5 Find the derivative of f −1 at the following points without solving
for f −1 .
(a) f (x) = x2 + 1 (for x ≥ 0) at the point (5, 2).
(b) f (x) = x2 − 2x − 3 (for x ≤ 1) at the point (12, −3).
Extra Problems for Personal Practice
Explain what each of the following means:
Problem 6
(a) sin−1 (x)
−1
(b) (sin(x))
(c) sin x−1
(d) f −1 (x)
(e) f (x−1 )
−1
(f) (f (x))
Problem 7 Suppose that f (x) is a differentiable function which is one-to-one.
Given the table of values below, find the value of (f −1 )0 (7).
x
f (x)
f 0 (x)
1
7
61
7
11
-17
11
1
71
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Breakout Session 13: Derivatives of inverse functions
Problem 8 Find the slope of the tangent line to the curve y = f −1 (x) at (4, 7)
2
if the slope of the tangent line to the curve y = f (x) at (7, 4) is .
3
5