====== How to Derive a Derivative ======

The derivative is derived from the original function.\\
The original function is input to the derivative function.\\
$f(x)$ is the original function, and $f\prime(x)$ is its derivative.

==== By Definition ====

$$ \frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{(x+h)-(x)} = \frac{f(x+h)-f(x)}{h}$$.


==== Power Rule ====
$$ \text{if }f(x) = x^r \hspace{45pt}  \text{ then }f\prime (x) = rx^{r-1}$$

So for example\\
if $f(x) = x^2$ then $f′(x^2) = 2x$, and\\
if $f(x) = x^3$ then $f\prime(x^3) = 3x^2$, and\\
if $f(x) = 2x^4$ then $f\prime(2x^4) = 8x^3$\\

For the linear function:
$$f(x) = ax + b$$

Rewrite it as:
$$f(x) = ax^1 + bx^0$$

Then apply the Power Rule.
$$f\prime(x) = a*1 + b*0 = a$$

==== Sum Rule ====
When two or more functions are summed together, take the derivative of each function separately, then add those two derivatives together.
$$f(x) = h(x) + g(x)$$
$$f\prime(x) = h\prime(x) + g\prime(x)$$

For example, in a polynomial function, treat each term as a separate function.  Derive each term separately using the Power Rule, then sum them together.
$$f(x) = 3x2 + 4x - 12$$
$$f\prime(x) = 6x + 4$$

==== Difference Rule ====
Same as the Sum Rule.

==== Product Rule ====

When two functions are multiplied together, take the derivative of each function separately, then multiply each times its opposite, and sum the pairs together.
$$f(x) = h(x) * g(x)$$

$$(h(x) * g\prime(x)) + ( h\prime(x) * g(x))$$


==== Quotient Rule ====
According to Salman Khan, the Quotient Rule is not needed,\\
because with algebra you can rewrite $\frac{1}{n^2}$ as $n^{-2}$ and use the Product Rule.

==== Chain Rule ====

When two functions are nested one inside the other, take the derivative of the inside function and the outside function separately and then multiply them together.
$$f(x) = g(h(x))$$
$$f\prime(x) = (h\prime(x)) * g\prime(h(x))$$

For example, consider this function.
$$f(x) = (2x+3)^5$$

The inside function is $h(x) = 2x+3$ \\
The outside function is $f(x) = g(h(x))^5$ \\

Calculate the derivatives separately.\\
inside: $h\prime (x) = 2$ \\
outside: $g\prime (x) = 5(h(x))^4 = 5(2x + 3)^4$

Multiply these two derivatives together, inside times outside. 
$$f\prime(x) = 2 * 5(2x + 3)^4 = 10(2x + 3)^4$$


==== References ====
  * http://tutorial.math.lamar.edu/Classes/CalcI/ChainRule.aspx
  * https://www.wyzant.com/resources/lessons/math/calculus/differentiation
  * Salman Khan youtubes https://youtu.be/h78GdGiRmpM