This is an illustration of derivatives of functions made from arithmetic with other functions. The differentiation rules of constant, constant times a function, sum of functions, difference of functions, product of functions and quotient of functions are all illustrated.[br]Only one check box can be checked and a new function is created as the sum, difference,product or quotient of the [math]f(x)[/math] and [math]g(x)[/math] functions.[br]The right side graph shows the derivative of the functions. The function values at the point [math]x_0[/math] are shown on each graph. The point can be moved on the left graph.[br][br]The derivatives of both functions and the new function are shown on the right graph.[br]The blue fine dashed curve if f(x), the purple dashed line is g(x) and the solid green line is h(x).

Tasks for students:[br][br]Click "Next f(x)" until [math]f(x)[/math] is a constant. Note the derivative of [math]f(x)[/math]. The constant rule for differentiation is [math](c)'=0[/math][br][br]From this setting check h(x)=f(x)*g(x) to get a constant times the function [math]g(x)[/math]. Vary the constant and note how the derivative changes. Try this with various g(x) functions.[br]The constant times a function differentiation rule is [math]\left( c \cdot g(x) \right)' = c \cdot g'(x) [/math][br][br]Check the h(x)=f(x)+g(x) to get the sum of function. Verify the rule for differentiating the sum of functions. [math]\left( f(x) + g(x) \right)' = f'(x) + g'(x)[/math][br][br]Check the h(x)=f(x)-g(x) to get the sum of function. Verify the rule for differentiating the difference of functions. [math]\left( f(x) - g(x) \right)' = f'(x) - g'(x)[/math][br][br]Check the h(x)=f(x)*g(x) to get the sum of function. Verify the rule for differentiating the product of functions. [math]\left( f(x) * g(x) \right)' = f'(x)g(x) + f(x) g'(x)[/math][br][br]Check the h(x)=f(x)/g(x) to get the sum of function. Verify the rule for differentiating the quotient of functions. [math]\left( f(x) /g(x) \right)' = \frac{ f'(x) g(x) - f(x) g'(x)}{\left( g(x) \right)^2}[/math]