Exercises

Exercises 1.4 Exercises

Let

f (x) = | x | = g (x)

and

h (x) = x^{2} .

Find the derivatives of

(f g) (x)

and

(h \circ f) (x)

x = 0 .

Let

f (x) = {\begin{cases} - 2 x & x \geq 0, \\ - 3 x & x < 0, \end{cases}

and

g (x) = {\begin{cases} 3 x & x \geq 0, \\ 2 x & x < 0. \end{cases}

For each for the following functions, determine whether it is differentiable at

x = 0

and if so, compute the derivative at

x = 0 .

A derivation is an operation

D

on functions (more general on anything that we can talk about sum and product) satisfying:

For example,

\frac{d}{d x}

is a derivation on differentiable functions. Let

D

be a derivation. Show that

D^{2} (f g) = (D^{2} f) g + 2 (D f) (D g) + f (D^{2} g) .

Find a general expression for

D^{n} (f g)

for

n \geq 2 .

Show that $D = x \frac{d}{d x}$ is a derivation on differentiable functions.
Show that $D c = 0$ for any constant $c$ and that $D x = x .$
Compute $D \frac{1}{1 - x} .$ Hint: Let $h = \frac{1}{1 - x} .$ Find $D h$ by applying $D$ to the relation $1 = h (1 - x) .$
Compute $D^{2} \frac{1}{1 - x} .$