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Michael Wandrey 9/19/06 Period 2
Derivative Portfolio – Part II
The power rule is usually expressed as (nx)^(n-1), where n is the power of the original term. This power rule can be used to find the derivative of any polynomial by applying it to all of the terms in the polynomial – therefore, x^2 becomes 2x, since 2 -1 = 1, and x^1 equals x. The 2 is also placed in front of the x in order to complete this application of the power rule. The power rule can be proven by applying the definition of the derivative to x^n, and solving to get the power rule.
Derivatives have properties, similar to limits. These properties are also similar to those of limits – there is the derivative of the sum (the derivative of a sum equals the sum of the derivatives, or if f(x) = g(x) + h(x), then f’(x) = g’(x) + h’(x) ), the derivative of a constant times a function (the derivative of a constant times a function is that constant times the derivative of the function, or, if k is a constant and f(x) = kg(x), then
f’(x) = kg’(x)), and the derivative of a constant (the derivative of a constant is equal to zero since they have no rate of change, or, if c is a constant and f(x) = c, then f’(x) = 0).
Michael Wandrey 9/24/06 Period 2
Derivative Portfolio III
We may prove that the derivative of the sine is the cosine, and the derivative of the cosine is the negative sine. We start by using the definition of the derivative to prove that if f(x) = sin x, then f’(x) = cos x. We also use the trigonometry identity –
Sin A – Sin B = 2 ( cos 0.5(A + B) sin 0.5(A - B))
Plugging this in to the definition of the derivative will yield f’(x) = cos x as h approaches zero.
Finding the derivative of cosine is made easier with an identity.
Cos x = Sin (pi/2 –x)
This would simplify to – sin x if we were to continue onward and take the derivative.
