Basics differential calculus

 

Calculation of the slope

Let f (x) be some function of x e.g. x²

the 1st derivative in general:

df (x) / dx = f ‘(x)

the second derivative in general: df (x) * df (x) / d (x) * d (x) or

df (x)²/d (x)² = f”

(x) f (x) = xexpn

f ‘(x) = n*xexp(n-1)’ ‘(x) = (n-1) nx x> 0

Our example f (x) = x², n = 2

first derivative

f ‘(x) = 2 * exp (2-1) = 2x

2nd derivative f ”(x)? f ‘(x) = 2x, n = 1

f ” (x) = 2 * 1 * exp (1-1) = 2 * 1 ° = 2

 

some special derivations

f(x)  = √x, f ‘(x) = 1/2√x

f(x) = 1/x, f ‘(x) = -1/x²

f(x) = lnx, f ‘(x) = 1/x

f(x) = sinx, f ‘(x) = cosx

f(x) = cosx, f ‘(x) = -sinx

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