Chapter 3.3 Practice Problems
EXPECTED SKILLS:
• Know how to compute the derivatives of exponential functions.
• Be able to compute the derivatives of the inverse trigonometric functions, specifically,
sin−1 x, cos−1 x, tan−1 x and sec−1 x.
• Know how to apply logarithmic differentiation to compute the derivatives of functions
of the form (f (x))g(x) , where f and g are non-constant functions of x.
PRACTICE PROBLEMS:
For problems 1-17, differentiate. In some cases it may be better to use logarithmic differentiation.
1. y = e6x
2. g(x) = xe2x
3. f (x) = 5x
2
4. y = ex cos x
2 (x−1)
5. g(x) = ex
6. f (x) =
7. f (x) =
1 − e2x
1 − ex
ln x
+ 3x
ex
8. f (x) = ln (ex + 5)
9. y = xx
2
2
10. f (x) = ecos
2x+sin2 2x
11. h(x) = exp
1
1 − ln x
x
12. f (x) = (ln x)e
2
13. g(x) = ecos
x−sin2 x
14. y = cos−1 (3x)
15. y = arcsin (x2 )
1
16. y =
arctan (ex )
x3
17. f (x) = arcsin (cos x)
18. Compute an equation of the line which is tangent to y = e3x at x = ln 2.
1
19. Compute an equation of the tangent line to f (x) = cos−1 x at x = .
2
20. Find all value(s) of x at which the tangent lines to f (x) = tan−1 (4x) are perpendicular
to the line which passes through (0, 1) and (2, 0).
21. Find a linear function T1 (x) = mx + b which satisfies both of the following conditions:
• T1 (x) has the same y-intercept as f (x) = e2x .
• T1 (x) has the same slope as f (x) = e2x at the y-intercept.
2
22. Compute an equation of the tangent line to the curve exy + y = x4 at (−1, 0).
23. The equation y 00 + 5y 0 − 6y = 0 is called a differential equation because it involves an
unknown function y and its derivatives. Find the value(s) of the constant A for which
y = eAx satisfies this equation.
√
−1
3
+ h − π3
sin
2
by interpreting the limit as the derivative of a func24. Evaluate lim
h→0
h
tion a particular point.
25. Consider the following two hyperbolic functions:
Hyperbolic Sine
Hyperbolic Cosine
ex − e−x
sinh x =
2
ex + e−x
cosh x =
2
(a) Compute lim sinh x
x→∞
(b) Compute lim sinh x
x→−∞
(c) Compute lim cosh x
x→∞
(d) Compute lim cosh x
x→−∞
2
(e) Show that cosh x − sinh2 x = 1
d
(f) Show that
(sinh x) = cosh x
dx
d
(g) Show that
(cosh x) = sinh x
dx
2