# 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