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After establishing the derivatives for essential core functions ( CCD.DCF) we consider how derivatives are found for more complicated but "elementary" functions by understanding how these functions are defined.

Elementary functions are defined from the core functions by using the arithmetic operations of addition, subtraction, multiplication, and division, along with composition of functions.
The rules for calculating the derivatives of elementary functions are usually referred to as the "derivative" calculus.

CCD.DP: The derivative of sums of functions: The Sum Rule.

$D_x (f +g) = D_x f +D_x g$ or $\frac {d (f +g)}{dx} = \frac {d f}{dx}+\frac {dg}{dx}$

$D_x (f - g) = D_x f - D_x g$ or $\frac { d(f -g)}{dx} = \frac {d f}{dx}-\frac {dg}{dx}$

CCD.DS: The derivative of constant (scalar) products of functions: The Constant(Scalar) Multiple Rule .


$D_x (cf ) = cD_x f $ or $\frac { d(cf )}{dx} = c\frac {d f}{dx}$

CCD.DS: The derivative of products of functions: The Product Rule.


$D_x (f \cdot g) = g \cdot D_x f + f \cdot D_x g$ or $\frac {d (f \cdot g)}{dx} = g \cdot \frac {d f}{dx}+ f \cdot \frac {dg}{dx}$

CCD.DS: The derivative of quotients of functions: The Quotient Rule.


$D_x ( \frac f g) = \frac {g \cdot D_x f - f \cdot D_x g} {g^2}$ or $\frac { d(\frac fg)}{dx} = \frac{g \cdot \frac {d f}{dx}- f \cdot \frac {dg}{dx}}{g^2}$

CCD.DS: The derivative of compositions of functions: The Chain Rule.


$D_x (f \circ g)) =( D_x f \circ g) \cdot D_x g$ or $\frac {d( f \circ g)}{dx} = \frac {d f}{dx}(g) \cdot \frac {dg}{dx}$

Dynamic visualization of the definitions of the derivative: