Table of Contents
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. $$f\prime(x) = 2 * 5(2x + 3)^4 = 10(2x + 3)^4$$