Log 0,5(2x-x^2) D(log)

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вопрос

To differentiate the function log base 0.5 of (2x - x^2), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative is f'(g(x)) * g'(x).

Let's break down the function: f(y) = log base 0.5(y), g(x) = 2x - x^2.

First, we find the derivative of f(y) = log base 0.5(y):
f'(y) = 1 / (y * ln(0.5))

Now, we find the derivative of g(x) = 2x - x^2:
g'(x) = 2 - 2x

Now, using the chain rule, the derivative of log base 0.5(2x - x^2) with respect to x is:

f'(g(x)) g'(x) = (1 / (2x - x^2) ln(0.5)) (2 - 2x)
= (1 / (2x - x^2) ln(0.5)) 2(1 - x)
= 2(1 - x) / (2x - x^2) ln(0.5)

Therefore, the derivative of log base 0.5(2x - x^2) with respect to x is 2(1 - x) / (2x - x^2) * ln(0.5).