Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 14
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41Worksheet # 10: The Derivative as Function, Product and Quotient
Rules
1. Comprehension check:
(a) True or false: If f (x) = g (x) for all x, then f = g?
(b) True or false: If f (x) = g(x) for all x, then f = g ?
(c) True or false: (f + g) = f + g
(d) True or false: (f g) = f g
(e) How is the number e defined?
(f) Are differentiable functions also continuous? Are continuous functions also differentiable?
2. Show by way of example that, in general,
d
df
dg
(f g) =
dx
dx
dx
and
df
d
f
dx
=
.
dg
dx
g
dx
3. Calculate the derivatives of the following functions in the two ways that are described.
(a) f (r) = r /3
i. using the constant multiple rule and the power rule
ii. using the quotient rule and the power rule
Which method should we prefer?
(b) f (x) = x
i. using the power rule
ii. using the product rule by considering the function as f (x) = x
x
(c) g(x) = (x + 1)(x
1)
i. first multiply out the factors and then use the power rule
ii. by using the product rule
4. State the quotient and product rule and be sure to include all necessary hypotheses.
5. Compute the first derivative of each of the following:
2x
(a) f (x) = (3x + x)e
(e) f (x) =
4 + x
x
(b) f (x) =
ax + b
x
1
(f) f (x) =
cx + d
e
(c) f (x) =
(x + 1)(x + 2)
2x
(g) f (x) =
x
x
1
(d) f (x) = (x + 2x + e )
(h) f (x) = (x
3)(2x + 1)(x + 5)
x
6. Let f (x) = (3x
1)e . For which x is the slope of the tangent line to f positive? Negative? Zero?
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