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Calculus/calc w/ trig differentiation

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Expert: - 5/6/2009


Question
given y=sinx(1-cosx) show dy/dx=(1+2cosx)(1-cosx)

Answer
This is a product rule.  The product rule states that if
y = f(x)g(x), then y' = f'(x)g(x) + g'(x)f(x).

Here, f(x) = sinx and g(x) = 1 - cosx.
That means that f'(x) = cos(x) and g'(x) = sin(x).

The derivative of y is then
y' = cosx*(1 - cosx) + sin²x.

Now remember that sin²x + cos²x = 1, so sin²x = 1 - cos²x.
That converts the equation
y' = cosx*(1 - cosx) + sin²x into
y' = cosx*(1 - cosx) + 1 - cos²x.

Next, remember that a² - b² = (a+b)(a-b).
Letting a be 1 and b be cosx, we can say
y' = cosx*(1 - cosx) + 1 - cos²x  is the same as
y' = cosx*(1 - cosx) + (1 + cosx)(1 - cosx).

Now note that there is a (1 - cosx) in both terms, so factoring gives
y' = (cosx + 1 + cosx)(1 - cosx).

Simply add the cosx and the cosx in the first set of parenthises,
and you're there.  

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