A PowerPoint presentation on the Derivative as a Function. Includes example problems on finding the derivative using the definition, Power Rule, examining graphs of f(x) and f'(x), and local linearity.
4. 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.
5. Prove that f (x) = x3 – 12x is differentiable.
Compute f ‘(x) and write the equation of the
tangent line at x = -3.
6. F ‘(x) = 3x2 – 12
Equation of tangent line at x = -3
y = 15x + 54
7. Calculate the derivative of y = x-2. Find the
domain of y and y’
8. Solution: y’ = -2x-3
Domain of y: {x| x ≠ 0}
Domain of y’ : {x| x ≠ 0}
The function is differentiable.
9. Another notation for writing the derivative:
Read “dy dx”
For the last example y = x-2, the solution could
have been written this way:
14. 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’
15. Find the points on the graph of
f(t) = t3 – 12t + 4 where the tangent line(s) is
horizontal.
17. How is the graph of f(x) = x3 – 12x related to
the graph of f’(x) = 3x2 – 12 ?
18. f(x) = x3 – 12 x Decreasing on (-2, 2)
Increasing on (2, ∞)
Increasing on (-∞, -2)
What happens
to f(x) at x = -2
f’(x) = 3x2 - 12 and x = 2??
Graph of f’(x) positive f’(x) is negative f’(x) is positive on (2, ∞)
on (-∞, -2) on (-2,2)
Zeros at -2, 2
19. Differentiability Implies Continuity
If f is differentiable at x = c, then f is continuous at
x = c.
20. Show that f(x) = |x| is continuous but not
differentiable at x = 0.
