the full rule for exponentials is: d/dx e^kx = k e^kx.
You have two ways to derive the differentiation rule of the natural logarithm. You could obviously evaluate the limit of the differencequotient, but there is a simpler way:
y(x)=ln(x) so
e^y = x, now differentiate wrt x:
d/dx (e^y) = d/dx x. Apply the chain rule to the LHS
e^y dy/dx = 1
dy/dx=1/e^y, and we know y=ln(x), so e^y = x and
dy/dx = 1/x
The chain rule can be confusing at first. Remember you have a function of the type
f(g(x)). eg: sin(x^2), where g=x^2 and f=sin(g); then differentiate the inner function to find g' -> 2x; differentiate the outer function: f'=cos(g); now the derivative is given by their product, so
g'(x)*f'(g) = 2x cos(g); and g= x^2, so finally: 2x cos(x²)