Calculus AB Homework 3.6 Derivatives of Inverses

This paragraph discusses the process of finding the derivatives of functions involving natural logarithms. The presenter uses the natural log rule, which states that the derivative of the natural log of a function is 1 over the function times the derivative of the function inside the log. Examples include finding the derivative of ln(e^(2x) + 1), simplifying it using the chain rule, and explaining the process step by step. The paragraph also covers other functions like ln(3 - 2x) and ln(√(5x - 2)), each with a detailed explanation of the derivative calculation.

The paragraph delves into the application of the chain rule for finding derivatives of composite functions. It provides a step-by-step breakdown of how to identify the inner and outer functions, and how to apply the chain rule to calculate the derivative of the natural log of a square root function. The explanation includes simplifying the derivative and using the long form of the chain rule to ensure clarity. The paragraph also touches on the derivative of a quotient of functions and the process of simplifying complex fractions.

This section focuses on the derivatives of inverse trigonometric functions, specifically arcsine and arctan. The presenter explains the derivative of arcsin(5x) by applying the chain rule and emphasizes the importance of the derivative of the inside function and the outside function. The explanation includes the simplification of the resulting expression. The paragraph also covers the derivative of arctan(1 - x), illustrating the process with a clear example and the final simplified form of the derivative.

The paragraph discusses the process of finding the derivative of the tangent inverse function and touches on the concept of differentiating inverse functions in general. It provides an example of finding the derivative of tan inverse (3/x) using the quotient rule and the power rule, leading to the simplified form of the derivative. The explanation also includes a three-step process for finding the derivative of an inverse function at a given point, which involves finding the original function's value, its derivative, and then taking the reciprocal of the derivative.

This section covers the process of finding the derivatives of functions and using them to determine the equations of tangent lines. It includes examples of finding the tangent line to the graph of an inverse function at a specific point, using the derivative to find the slope of the tangent line and the function value to find the point of tangency. The explanation involves setting up the equation of the tangent line and simplifying it to the final form. The paragraph also discusses finding the equation of the normal line to a graph, which involves using the negative reciprocal of the slope of the tangent line.

The paragraph explains the method of finding the value of an inverse function at a specific point and then using that to determine the derivative of the inverse function. It provides a detailed example of finding F inverse prime of 2, which involves solving for the original function's value, evaluating the derivative at that point, and taking the reciprocal. The explanation also includes finding the equation of a tangent line to the graph of an inverse function at a given x-value, using the derivative and the function value to establish the slope and point of tangency.

This section discusses the process of finding the derivatives of composite functions that involve inverse functions. It provides examples of setting up equations to find the value of the inverse function at a specific point and then using that to find the derivative of the inverse function. The explanation includes the use of the chain rule to find the derivative of the composite function and the process of simplifying the resulting expression to find the derivative at a given point.

The paragraph covers the process of finding derivatives for more complex functions, including those involving exponential and logarithmic components. It provides a detailed example of finding the derivative of a function that combines an exponential term with a logarithmic term, using the product rule and the chain rule. The explanation includes simplifying the derivative and finding the reciprocal to get the final answer.

This final paragraph wraps up the series of derivative calculations with a few more examples of finding derivatives of complex functions. It includes functions with exponential and inverse terms, where the presenter demonstrates the process of taking derivatives using the quotient rule and the chain rule. The explanation involves simplifying the derivatives and finding the reciprocal to get the final answers, showcasing the application of calculus concepts in solving these problems.