MATH 131-505 Spring 2015
3.1
c
Wen
Liu
Chapter 3 Differentiation Rules
3.1 Derivatives of Polynomials and Exponential Functions
Theorem:
1.
d
(c) = 0 for any constant c.
dx
2. The Power Rule:
3.
d n
(x ) = nxn−1 for any real number n.
dx
d x
(e ) = ex
dx
4. If f 0 (x) and g 0 (x) exist, then
d
d
(cf (x)) = c (f (x)).
dx
dx
d
d
d
(b) The Sum/Difference Rule:
(f (x) ± g(x)) =
(f (x)) ±
(g(x))
dx
dx
dx
(a) The Constant Multiple Rule:
To find f 0 (a) (i.e., the derivative of f (x) at x = a):
1. find f 0 (x)
2. substitute x = a into f 0 (x).
Warning:
constant.
If you compute f (a) and then find the derivative, you will always get 0 since f (a) is a
Note: π, e are constants.
Examples:
1. Differentiate each of the following functions.
(a) f (x) = −1.052
√
3
(b) g(x) = a x2 for some constant a 6= 0
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MATH 131-505 Spring 2015
3.1
c
Wen
Liu
√
(c) h(x) = (x − 4)(5 x + 15)
(d) l(x) = −
5
2x2 + 3
−3
√
+
x
+
x2
x
2. (p. 178) The equation of motion of a particle is s = 2t3 − 5t2 + 3t + 4, where s is measured in
centimeters and t in seconds.
(a) Find the velocity and acceleration as functions of t.
(b) Find the acceleration after 2 seconds.
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MATH 131-505 Spring 2015
3.1
c
Wen
Liu
3. If f (x) = ex − 3x2 , find f 00 (0).
4. (p. 181) At what point on the curve y = ex is the tangent line parallel to the line y = 2x?
The normal line to a curve C at a point P is the line through P that is perpendicular to the tangent
line at P .
Examples:
√
5. (p. 176) Find equations of the tangent line and normal line to the curve y = x x at the point
(1, 1). Illustrate by graphing the curve and these lines.
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MATH 131-505 Spring 2015
3.1
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Wen
Liu
6. If y = 3x3 + 5x2 , where the tangent is horizontal?
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