91Ó°ÊÓ

The chain rule is a fundamental concept in calculus used to differentiate composite functions. When a function is composed of two or more functions, the derivative can be found using the chain rule.

• It is particularly useful when dealing with functions of the form: \( g(f(x)) \).
• In essence, the chain rule helps calculate how a change in one variable causes a change in another variable through intermediate dependencies.

For example, if you have a function \( y = e^{\sin x} \), you apply the chain rule to differentiate. Here, \( \sin x \) is the inner function \( f(x) \) and \( e^u \) is the outer function \( g(u) \) where \( u = \sin x \).To find the derivative:1. Differentiate the outer function with respect to the inner function: The derivative of \( e^u \) with respect to \( u \) is \( e^u \).2. Differentiate the inner function with respect to \( x \): The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \).3. Multiply these derivatives together: \( \frac{dy}{dx} = e^{\sin x} \cdot \cos x \).This powerful technique simplifies the process of dealing with complex compositions.

Logarithmic functions are the inverse of exponential functions, playing a crucial role in mathematics, especially when dealing with growth and decay problems.

• A logarithm \( \ln y \) expresses the power to which a base (in natural logs, this base is \( e \)) must be raised to produce a given number \( y \).
• This is expressed as \( \ln y = \log_e y \).

In the original exercise, the equation \( \sin x = \ln y \) was given. To solve for \( y \), you "undo" the logarithm by turning it into an exponential equation:- Using the property of logarithms that states \( y = e^u \) where \( u = \ln y \), yields \( y = e^{\sin x} \).When dealing with derivatives and logarithms, it's beneficial to understand this conversion between logarithmic and exponential forms. This understanding is indispensable in finding derivatives where logarithms are present.

Trigonometric functions, including sine, cosine, and others, describe the relationships between the angles and sides of triangles.

• They are periodic and provide a way to model oscillatory phenomena such as waves.
• The basic trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)).

In the problem at hand, the function \( \sin x \) is involved. Knowing the derivatives of basic trigonometric functions is crucial in calculus:- The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \).- Understanding this helps solve the given problem by applying this knowledge within the chain rule.This is particularly useful when differentiating composite functions that involve trigonometric elements, as it simplifies the identification of derivatives. Comprehending and being comfortable with the derivatives of sine and cosine aids in solving a wide variety of calculus problems.