Differential calculus


THEOREM   

If \(f(x)\) is differentiable at \(x = a\), then \(f(x)\) is continuous at \(x = a\).

Proof available

THEOREM    \((f(x) + g(x))' = f'(x) + g'(x)\)

Proof available

THEOREM    \((f(x) - g(x))' = f'(x) - g'(x)\)

Proof available

THEOREM    \((cf(x))' = cf'(x)\)

Proof available

THEOREM    \(\dfrac{d}{dx}(c) = 0\)

Proof available

THEOREM    \(\dfrac{d}{dx}\left(x^n\right) = nx^{n-1}\)

Proof available

THEOREM    \((fg)' = f'g + fg'\)

Proof available

THEOREM    \(\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}\)

Proof available

THEOREM   

If \(f(x)\) and \(g(x)\) are both differentiable functions, and we define \(F(x) = (f \circ g)(x)\), then the derivative of \(F(x)\) is \(F'(x) = f'(g(x))g'(x)\).

Proof available