Question

Prove that the following functions are differentiable, and find their derivatives at an arbitrary point:

a. f(x) = sin(x)
b. g(x) = eˣ
c. h(x) = ln(x)
d. k(x) = 1/x

Answer 1

These results are based on standard rules of calculus and the properties of elementary functions to evaluate these expressions at the given point. If you have a specific point in mind, please provide it so I can assist you further.

Step-by-step explanation:

Let's find the derivatives for each function:

a. f(x)=sin(x)

The derivative of sin(x) is cos(x).

b. g(x)=e x

The derivative of ex is ex.

c. h(x)=ln(x)

The derivative of ln(x) is 1/x.

d. k(x)= 1/x.

The derivative of 1/x is - 1/x 2.

So, the derivatives are:

f ′(x)=cos(x)

g ′(x)=e x

h ′(x)= x1

k ′(x)= − 1/x2

To find the derivatives at an arbitrary point, you need to evaluate these expressions at the given point.

These results are based on standard rules of calculus and the properties of elementary functions.