Suppose that the functions ƒ and g are defined throughout an openinterval containing the point x0, that ƒ is differentiable at x0, thatƒ(x0) = 0, and that g is continuous at x0. Show that the productƒg is differentiable at x0. This process shows, for example, thatalthough | x | is not differentiable at x = 0, the product x | x | isdifferentiable at x = 0.
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Suppose that the functions ƒ and g are defined throughout an open
interval containing the point x0, that ƒ is differentiable at x0, that
ƒ(x0) = 0, and that g is continuous at x0. Show that the product
ƒg is differentiable at x0. This process shows, for example, that
although | x | is not differentiable at x = 0, the product x | x | is
differentiable at x = 0.
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