Holooly Plus Logo
Holooly Plus Logo

Question 7.T.3: Let the functions f and g be defined on the real intervals I...

Let the functions f and g be defined on the real intervals I and J, respectively, such that f(I) ⊆ J. If f is differentiable at c ∈ I, and g is differentiable at f(c), then the composed function

g \circ f : I → \mathbb{R}

is differentiable at c, where its derivative is given by

(g \circ f )^{′}(c) = g^{′}(f (c)) · f^{′}(c).

Before we prove this theorem, it is worth pointing out a quick, but false, “proof” which begins with writing

\frac{(g  \circ f  )(x) − (g  \circ  f )(c)}{x − c} = \frac{g(f(x)) − g(f(c))}{f (x) − f (c)} · \frac{f(x) − f(c)}{x − c},

and then taking the limit as x → c. But this equation is valid only if f(x)−f(c) ≠ 0. Though we know that x−c ≠ 0, we cannot assume that f(x)−f(c) ≠ 0. The following proof covers the case when f(x)−f(c) = 0.

The blue check mark means that this solution has been answered and checked by an expert. This guarantees that the final answer is accurate.
Learn more on how we answer questions.

Let d = f(c) and define the function φ on J by

\varphi(y)= \begin{cases} \frac{g(y) − g(d)}{y − d}, &y \neq d \\ g^{′}(d), & y = d.\end{cases}

Observe that

\underset{y→d}{\lim}  \varphi(y) = g^{′}(d) = \varphi(d),

and that g(y) − g(d) = (y − d)φ(y) for all y ∈ J , including y = d.

Now let x ∈ I\{c} and y = f(x).

(g ◦ f )(x) − (g ◦ f )(c) = g(y) − g(d)

= (y − d)φ(y)

⇒ \frac{(g  \circ  f )(x) − (g  \circ  f )(c)}{x − c} = \frac{y − d}{x − c}\varphi(y)

=\frac{f(x) − f(c)}{x − c}\varphi(y).

As x → c,

\frac{f(x) − f(c)}{x − c}→ f^{′}(c),

and y → d, by the continuity of f, hence \varphi(y) → g^{′}(d) by (7.1). Thus

\underset{x→c}{\lim}\frac{(g  \circ  f )(x) − (g  \circ  f )(c)}{x − c} = g^{′}(d) · f^{′}(c).

Related Answered Questions

Question: 7.T.18

Let the function f and all its derivatives up to order n be continuous on [a, b], and suppose f^(n) is differentiable at the point x0 ∈ [a, b]. If x ∈ [a, b] then f(x) = f(x0) + f′(x0)/1! (x − x0) + f′′(x0)/2! (x − x0)^2 + · · · + f^(n)(x0)/n! (x − x0)^n + f^(n+1)(x0)/(n + 1)! (x − x0)^n+1 + En(x), ...

Verified Answer:

Let p be the polynomial p(x) = f(x_{0})+  \...
Question: 7.T.16

Let the functions f and g be differentiable on (a, b) and g′(x) ≠ 0 for any x ∈ (a, b). If limx→a^+ g(x) = limx→a^+ f(x) = ∞ and limx→a^+ f′(x)/g′(x) exists in R, then limx→a^+ f(x)/g(x) = lim^x→a+ f′(x)/g′(x). ...

Verified Answer:

We shall prove the theorem in the case when [latex...
Question: 7.T.15

Let the functions f and g be differentiable on [a, ∞) and suppose limx→∞ f(x) = limx→∞ g(x) = 0, g′(x) ≠ 0 for all x > a. If limx→∞ f′(x)/g′(x) exists in R, then limx→∞ f(x)/g(x) = limx→∞ f′(x)/g′(x). ...

Verified Answer:

Clearly, we can take a > 0. Define the function...
Question: 7.T.12

(Darboux) Let the function f : [a, b] → R be differentiable. If λ is a number between f′(a) and f′(b), then there is a point c ∈ (a, b) such that f′(c) = λ. ...

Verified Answer:

By using −f instead of f if necessary, we may assu...
Question: 7.T.11

(Inverse Function Theorem) Let f be a differentiable function whose derivative is continuous on the open interval I. If f′(b) ≠ 0 for some b ∈ I, then there is an open interval J ⊆ I which contains b such that f|J is injective, and the function (f|J )^−1 is differentiable on f(J), where its ...

Verified Answer:

Since f^{′} is continuous at b, by ...
Question: 7.T.10

Let c be a critical point and a point of continuity for the function f : D → R. (i) If there is an open interval I ⊆ D, which contains c, such that f′(x) < 0 for all x ∈ I, x 0 for all x ∈ I, x > c, then f(c) is a local minimum for f. (ii) If there is an open interval I ⊆ D, which ...

Verified Answer:

Figure 7.7 illustrates the three cases of the theo...
Question: 7.T.8

Let f : [a, b] → R be a continuous function which is differentiable on (a, b). (i) If f′(x) = 0 for all x ∈ (a, b), f is constant on [a, b]. (ii) If, for all x ∈ (a, b), f′(x) ≠ 0, f is injective on [a, b]. ...

Verified Answer:

(i) Take any two points x_{1}, x_{2} ∈ [a, ...
Question: 7.T.14

(L’Hôpital’s Rule) Let the functions f and g be continuous on a real interval I and differentiable on I\{c}, where c is a point in I. If (i) g′(x) ≠ 0 for all x ∈ I\{c}, (ii) f(c) = g(c) = 0, (iii) limx→c f′(x)/g′(x) exists in R = R ∪ {−∞, ∞}, then limx→c f(x)/g(x) = limx→c f′(x)/g′(x). ...

Verified Answer:

Let (x_{n}) be any sequence in I su...
Question: 7.T.13

(Cauchy’s Mean Value Theorem) If the functions f and g are continuous on [a, b] and differentiable on (a, b), then there is a point c ∈ (a, b) such that [f(b) − f(a)]g′(c) = [g(b) − g(a)]f′(c). ...

Verified Answer:

Let h be the function defined on [a, b] by h(x) = ...
Question: 7.T.9

Let f : [a, b] → R be a continuous function which is differentiable on (a, b). (i) If f′(x) ≥ 0 for all x ∈ (a, b), f is increasing on [a, b]. (ii) If f′(x) ≤ 0 for all x ∈ (a, b), f is decreasing on [a, b]. (iii) If f′(x) > 0 for all x ∈ (a, b), f is strictly increasing on [a, b]. (iv) If ...

Verified Answer:

Suppose x_{1}, x_{2} ∈ [a, b] and [...