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3.2 Derivative as a Function
AP Calculus
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Derivative at a Point x = a
Derivative f’ (a) for specific values of a
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Derivative as a Function
Function f’ (x)
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Domain and Differentiability
Domain of f’ (x) is all values of x in domain of
f (x) for which the limit exists. F’ (x) is differentiable on (a, b) if f
‘(x) exists for all x in (a, b). If f’ (x) exists for all x, then f (x) is
differentiable.
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Example 1
Prove that f (x) = x3 – 12x is differentiable. Compute f ‘(x) and write the equation of the tangent line at x = -3.
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Solution
F ‘(x) = 3x2 – 12 Equation of tangent line at x = -3
y = 15x + 54
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Example 2
Calculate the derivative of y = x-2. Find the domain of y and y’
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Example 2
Solution: y’ = -2x-3
Domain of y: {x| x ≠ 0} Domain of y’ : {x| x ≠ 0} The function is differentiable.
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Leibniz Notation
Another notation for writing the derivative:
Read “dy dx” For the last example y = x-2, the
solution could have been written this way:
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Theorem 1: Power Rule
For all exponents n,
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Power Rule Examples
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Example 3
Calculate the derivative of the function below
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Example 3
Solution:
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Theorem 2
Assume that f and g are differentiable functions.
Sum Rule: the function f + g is differentiable (f + g)’ = f’ + g’
Constant Multiple Rule: For any constant c, cf is differentiable and
(cf)’ = cf’
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Example 4
Find the points on the graph of f(t) = t3 – 12t + 4 where the tangent line(s) is horizontal.
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Example 4
Solution:
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Graphical Insight
How is the graph of f(x) = x3 – 12x related to the graph of f’(x) = 3x2 – 12 ?
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f(x) = x3 – 12 x
f’(x) = 3x2 - 12
Increasing on (-∞, -2)
Graph of f’(x) positive on (-∞, -2)
Zeros at -2, 2
Decreasing on (-2, 2)
f’(x) is negative on (-2,2)
Increasing on (2, ∞)
f’(x) is positive on (2, ∞)
What happens to f(x) at x = -2 and x = 2??
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Theorem 3
Differentiability Implies Continuity If f is differentiable at x = c, then f is
continuous at x = c.
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Example 5
Show that f(x) = |x| is continuous but not differentiable at x = 0.
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Example 5 - Solution
The function is continuous at x = 0 because
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Example 5 - Solution
The one-sided limits are not equal:
The function is not differentiable at x = 0
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Graphical Insight
Local Linearity f(x) = x3 – 12x
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Graphical Insight
g(x) = |x|
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Example 6
Show that f(x) = x 1/3 is not differentiable at x = 0.
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Example 6 - Solution
f’(0) =
The limit at x = 0 is infinite
The slope of the tangent line is infinite – vertical tangent line