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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.10.38
Chapter 3, Problem 3.10.38

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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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Intro to the Chain Rule

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.
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Derivatives of Other Inverse Trigonometric Functions

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).
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Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question

Find the following higher-order derivatives.

dn/dxn (2x)

Textbook Question

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x) = 1/2x+8; (10,4)

Textbook Question

73–78. {Use of Tech} Normal lines A normal line at a point P on a curve passes through P and is perpendicular to the line tangent to the curve at P (see figure). Use the following equations and graphs to determine an equation of the normal line at the given point. Illustrate your work by graphing the curve with the normal line. <IMAGE>


Exercise 46

Textbook Question

Find the derivative of the following functions.

y = cot x / (1 + csc x)

Textbook Question

Consider the line f(x)=mx+b, where m and b are constants. Show that f′(x)=m for all x. Interpret this result.

Textbook Question

Find the slope of the line tangent to the graph of y = tan^−1 x at x= −2.