【Calculus & Algebra】Limits & Derivatives

Basics of Derivatives

Derivative:

Let $y=f(x)$, then $$f'(x) = \frac{dy}{dx} = lim_{\Delta x \to 0} \frac{f(x+\Delta x) - f(x)}{\Delta x}$$

 

 

The Product Rule:

 

If $u=u(x)$ and v=v(x) and their respective derivatives exist, 

 

$$\frac{d(uv)}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}, (uv)' = u'v + uv'$$

 

 

The quotient rule:

$$\frac{d}{dx}(\frac{u}{v}) = \cfrac{(v\frac{du}{dx}-u \frac{dv}{dx})}{v^2}, (\frac{u}{v})' = \frac{u'v-uv'}{\Delta x}$$

 

 

The chain rule:

If $y=f(u(x))$ and $u=u(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$

 

 

The generalized power rule:

$$\frac{dy^n}{dx} = ny^{n-1} \frac{dy}{dx}$$ for $$\forall n \neq 0 $$

 

 

Other Useful Eqquations:

$$a^x = e^{x \ln a} $$

 

$$\ln(ab) = \ln a \ln b $$

 

$$\lim{x \to \infty} \frac{\sin x}{x} = 1$$

 

$$\lim{x \to \infty} (1+x)^k = 1 + kx$$ for any $k$

 

$$\lim{\ln x \to x^r} = 0 $$ for any $ r > 0$

 

$$\lim{x \to \infty} x^r e^{-x} = 0$$ for any $r$

 

$$\frac{d}{dx} e^u = e^u \frac{du}{dx}$$

 

$$\frac{da^u}{dx} = (a^u \ln a) \frac{du}{dx}$$

 

$$\frac{d}{dx} \ln u = \frac{1}{u} \frac{du}{dx} = \frac{u'}{u}$$