Problem 1

Compute the derivative of the following functions directly from the definition

3. f(x) = x³.
4. f(x) = ^p(x).
5. f(x) = x.
6. f(x) = c with some constant c.

Problem 2

Compute the derivatives of the following functions

1. where b is a constant
2. g(t) = cos(ωt + φ) + sin(ωt + φ) where ω and φ are constants
3. h(s) = cos(s² + s) + sin(s/2)
4. j(x) = ln(x^a2 + x^−a2) where a is a constant

Note: You can use (lnx)⁰ = 1/x from the lecture

1. e) k(x) = ln(x^a + b^x) where a and b are constants
2. l(x) = x² exp(−x²)
3. m(x) = x^x2

Note for e) and g): You cannot directly work with something of the form a^x (with some a) but only with something of the form e^cx (with some c). Transform the function accordingly before differentiation.

Jacobs University of Bremen

Problem 3

Use the definition of the derivative, , to show that the function f(x) = |x| is not differentiable at x = 0.