Derivatives

Overview

Calculus is mainly about calculating the area under a curve. We set a starting point on the X-axis and set $x$ as the moving endpoint on the X-axis.

If the curve is $f(x)$, and the area under the curve is $A(x)$. We have $dA$ = $f(x)dx$, where $dx$ is a very tiny length on the X-axis, $f(x)dx$ is the area of the very thin rectangle at $x$. $dA$ means the very small change of function $A$ from $x$ to $x+dx$. From $dA=f(x)dx$, we have $dA/dx = f(x)$. We call $f(x)$ the derivative of function $A$.

To calculate the derivative of A: $\frac {dA}{dx}$ as $\frac{A(x+dx)-A(x)}{dx}$. By some transformation we can get the result. For example, $A(x) = x^2$, $\frac{dA}{dx} = \frac{(x+dx)^2-x^2}{dx} = \frac{2xdx + d^2x}{dx}= 2x + dx$. Since $dx$ is very small, so $\frac{dA}{dx} = 2x$.

Don't think $dx$ as infinitely small, but think of it as a comparatively small value.

Derivative Rules

Three rules of derivative.

• Constant rule: $f'(x) = 0$

• Sum rule: $(\alpha f(x)+ \beta g(x))' = \alpha f'(x) + \beta g'(x)$

• Product rule: $(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)$

• Quotient rule: $\left({\frac {f(x)}{g(x)}}\right)'={\frac {f'(x)g(x)-f(x)g'(x)}{g^{2}(x)}}$

• Chain rule: $(f(g(x)))' = f'(g(x))g'(x)$

Mathimatical Constant

A simple definition of the mathimatical constant $e$ is $(e^x)' = e^x$.

Implicit Differentiation

For example, $f(x, y)=g(x, y)$. From this equation, we cannot directly see a form of $y=...$. We cannot directly calculate the derivative. What we know is this equation defines a curve (or some dots or something) in the plane.

We can conclude $f'(x, y)=g'(x, y)$. Two questions needs to be answered.

• Is $f'(x, y)$ the derivative of $x$ or $y$? It doesn't matter, since $dy$ can be seen as a function of $dx$, and vice versa.

• Why they still equal when calculate derivative? The curve defined by $f(x, y)$=$g(x, y)$ constrain $x$ and $y$ to have a certain relationship. It might be the intervals, where $f$ and $g$ overlap. We don't need to consider single points overlaps since there is no derivative if they are scattered single points. Therefore, the equation is a definition of some curve in some intervals. As long as point $(x, y)$ is on the curve, we have $f(x, y)=g(x, y)$. Therefore, when we move a little bit $f$ changes by $df$, which is $f + df$, on the right of the equation is $g + dg$. Since $f=g$ and $f +df = g + dg$, $df =dg$.