Derivatives of Combinations of Functions

Sum or Difference of Functions:
If $f(x)$ and $g(x)$ are differentiable, then: $$$$
$(f + g)'(x) = f'(x) + g'(x)$$(f – g)'(x) = f'(x) – g'(x)$
This means if you want to find the rate of change of two functions added or subtracted, you can find the derivative of each and then add or subtract them.

Scalar Multiplication:
If $k$ is a constant and $f(x)$ is differentiable: $$$$
$(kf)'(x) = k f'(x)$
This means if you multiply a function by a number, you can find the derivative by multiplying that number by the derivative of the function.

Product Rule:
If $f(x)$ and $g(x)$ are differentiable, then: $$$$
$(fg)'(x) = f'(x)g(x) + f(x)g'(x)$
This is called the product rule, which is used when two functions are multiplied together.

Quotient Rule:
If $f(x)$ and $g(x)$ are differentiable, then the derivative of the quotient is: $${}^{}$$
$left( frac{f}{g} right)'(x) = frac{f'(x)g(x) – f(x)g'(x)}{g(x)^2}$
This is called the quotient rule, used when one function is divided by another.