To differentiate the function f(x) = sin³(ln(x)), we will apply the chain rule and the product rule of differentiation. Let’s break this down step by step.
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First, identify the outer and inner functions. Here, the outer function is sin³(u) where u = ln(x).
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To differentiate the outer function, we know that the derivative of sin³(u) can be expressed using the chain rule:
f'(x) = 3sin²(u) * cos(u) * u’, where u’ = d(ln(x))/dx.
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Now, we need to differentiate u = ln(x). The derivative of ln(x) is:
u’ = 1/x.
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Substituting back to our expression for f'(x):
f'(x) = 3sin²(ln(x)) * cos(ln(x)) * (1/x).
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Finally, we can simplify this into a more readable format:
f'(x) = (3sin²(ln(x)) * cos(ln(x))) / x.
In summary, the derivative of the function f(x) = sin³(ln(x)) is:
f'(x) = (3sin²(ln(x)) * cos(ln(x))) / x.