Calculus 1 The Definition of the Derivative
Assignment
1. Find the equation of the tangent line to the graph of f (x) = x2 + 2 at x = 3.
2. Find the derivative of f (x) =
3
by using the defintion of the derivative.
x
3. Show that f (x) = |x + 3| is not differentiable at x = −3.
4. Is f (x) =
|x|
differentiable at x = 0? Justify your answer.
x
Solutions
1. Use the definition of the derivative to arrive at f 0 (x) = 2x. The slope of the tangent line to
the graph of f at x = 3 is f 0 (3) = 6. Now use the point-slope equation of a line y−y1 = m(x−x1 )
to arrive at y = 6x − 7.
2. f 0 (x) = −
3
x2
3. We’ll use this version of the definition: f 0 (c) = lim
x→c
f (x) − f (c)
.
x−c
lim
|x + 3| − | − 3 + 3|
|x + 3|
−(x + 3)
= lim
= lim
= lim (−1) = −1.
−
−
x − (−3)
x+3
x+3
x→−3
x→−3
x→−3−
lim
|x + 3| − | − 3 + 3|
|x + 3|
x+3
= lim
= lim
= lim 1 = 1.
+
+
x − (−3)
x+3
x→−3
x→−3 x + 3
x→−3+
x→−3−
x→−3+
Since the one sided limits are not the same, lim
x→−3
|x + 3| − | − 3 + 3|
DNE, hence f (x) = |x + 3|
x − (−3)
is not differentiable at x = −3.
4. This function is not continuous at zero and so it cannot be differentiable there.