Section 4.2: Basic Rules of Differentiation
Theorem: (Differentiation Rules)
Suppose that c is a constant and f and g are differentiable functions.
1. (Constant Rule)
If f (x) = c is a constant function, then f 0 (x) = 0.
2. (Power Rule)
If f (x) = xn , then f 0 (x) = nxn−1 .
3. (Constant Multiple Rule)
d
[cf (x)] = cf 0 (x).
dx
4. (Sum/Difference Rule)
d
[f (x) ± g(x)] = f 0 (x) ± g 0 (x).
dx
Example: Differentiate each function.
(a) f (x) = x8 + 6x7 − 18x2 + 2x + 5
(b) g(x) = (x2 + 1)(2x − 7)
1
(c) h(x) =
√
1
x− √
x
(d) f (t) =
√
2
3 2
t +
t
(e) R(x) =
x2 + 4x + 3
√
x
2
Example: Consider the logistic growth equation
dN
= rN
dt
N
1−
= f (N ),
K
where r and K are positive constants. Differentiate f (N ) with respect to N .
Example: Find an equation of the tangent line to the graph of y = 3x2 − 4x + 7 at x = 2.
Example: Find an equation of the normal line to the graph of y = 1 − 3x2 at x = −2.
3
Example: Find the points on the curve y = 2x3 + 3x2 − 12x + 1 where the tangent line is
horizontal.
Example: Find the points on the curve y = 2x3 − 4x + 1 where the tangent line is parallel
to the line y = 2x + 1.
4
Example: Show that there are two tangent lines to the parabola y = x2 that pass through
the point (0, −4). Find the equation of both tangent lines.
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