Evaluate the derivative of the following functions.
f(x) = sin(tan^-1 (ln x))

검증된 단계별 안내

First, identify the outermost function in the composition. Here, the function is f(x) = sin(u), where u = tan^(-1)(ln(x)).

Apply the chain rule for derivatives, which states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this case, differentiate sin(u) with respect to u, which gives cos(u).

Next, differentiate the inner function u = tan^(-1)(ln(x)) with respect to x. Start by differentiating tan^(-1)(v) with respect to v, which is 1/(1+v^2). Here, v = ln(x).

Now, differentiate ln(x) with respect to x, which is 1/x. Combine this with the previous step to find the derivative of u with respect to x: (1/(1+(ln(x))^2)) * (1/x).

Finally, combine the results using the chain rule: f'(x) = cos(tan^(-1)(ln(x))) * (1/(1+(ln(x))^2)) * (1/x). This is the derivative of the given function.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Chain Rule

The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. It states that if a function y is composed of two functions u and v, then the derivative of y with respect to x can be found by multiplying the derivative of y with respect to u by the derivative of u with respect to x. This rule is essential for evaluating derivatives of functions like f(x) = sin(tan<sup>-1</sup>(ln x)), where multiple functions are nested.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan<sup>-1</sup>(x), are the functions that reverse the action of the standard trigonometric functions. They are crucial in calculus for finding angles when given a ratio of sides in a right triangle. Understanding how to differentiate these functions is vital when evaluating derivatives involving them, as they have specific derivative formulas that must be applied correctly.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is an important function in calculus, particularly in differentiation and integration. The derivative of ln(x) is 1/x, which is a key component when differentiating functions that include the natural logarithm, such as ln(x) in the given function f(x).

Evaluate the derivative of the following functions.f(x) = sin(tan-1 (ln x))