Section 3.2: Differentiation Formulas
Theorem: (Differentiation Formulas)
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
(b) g(x) = (x2 + 1)(2x − 7)
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(c) h(x) =
√
1
x− √
x
x2 + 4x + 3
√
(d) R(x) =
x
(e) f (t) =
√
√
3 2
t + 2 t3
Example: Find an equation of the tangent line to the graph of y = x +
2
4
at (1, 5).
x
Example: Find the points on the curve y = 2x3 − 3x2 − 12x + 1 for which the tangent line
is horizontal.
Example: For what values of a and b is the line 2x + y = b tangent to the parabola y = ax2
when x = 2?
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Theorem: (Product Rule)
Suppose that f and g are differentiable functions. Then
d
[f (x)g(x)] = f 0 (x)g(x) + f (x)g 0 (x).
dx
Example: Differentiate each function.
(a) f (x) = (x2 + x + 1)(x2 + 2).
(b) f (x) = (2x2 − 5x + 1)2
(c) f (x) = (2x + 1)(4 − x2 )(1 + x2 )
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Theorem: (Quotient Rule)
Suppose that f and g are differentiable functions and g(x) 6= 0. Then
d f (x)
f 0 (x)g(x) − f (x)g 0 (x)
=
.
dx g(x)
[g(x)]2
Example: Differentiate each function.
(a) f (x) =
x5
x3 − 2
(b) g(x) =
x4 + 2x − 1
5x2 − 2x + 1
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Example: Find an equation of the tangent line to the curve y =
Example: Find the values of m and b that make
f (x) =
x2
if x ≤ 2
mx + b if x > 2
differentiable everywhere.
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x2 + 3
at x = −2.
x3 + 5
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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