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Chapter 2: Problem 30

Find an equation of the tangent line to $$f(x)=2 \ln x^{3}$$

Short Answer

Expert verified
The equation of the tangent line to the function \(f(x)=2 \ln x^{3}\) at any arbitrary point x can be represented by \(y-2\ln(x^{3})=6/x*(x-x_{1}).\)

Step by step solution

01

Compute the derivative of the function

The derivative of \(f(x) = 2 \ln x^{3}\) can be computed using the chain rule of differentiation. The rule states that if a function y = f(g(x)) is the composite of the functions f and g, then its derivative dy/dx = f'(g(x)) 路 g'(x). In this case, x^3 is g(x) and 2ln(x) is f(x). Therefore, \(f'(x) = 2 * (1/x^{3}) * 3x^{2} = 6/x.\)
02

Substitute value of x in f'(x)

The slope of the tangent line is equal to the value of the derivative at that point. To find equation of the tangent line, we need one more point, which is given by f(x). f(x)=2ln\(x^{3}\). What is left now is to choose a specific x in both of these formulas and you get a slope and a point for the tangent line.
03

Apply point slope form to find equation

The equation of a line in point-slope form is given by \(y-y_{1}=m(x-x_{1})\), where m is the slope of the line and \((x_{1},y_{1})\) is a point on the line. By substituting the calculated slope and point into this formula, the equation of the tangent line can be generated.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Chain Rule of Differentiation
Understanding the chain rule of differentiation is crucial for tackling calculus problems involving composite functions. It's like having a flashlight in a dark room 鈥� it helps you see and differentiate functions that are nested within each other.

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Most popular questions from this chapter

Explain why it is not valid to use the Mean Value Theorem. When the hypotheses are not true, the theorem does not tell you anything about the truth of the conclusion. In three of the four cases, show that there is no value of \(c\) that makes the conclusion of the theorem true. In the fourth case, find the value of \(c .\) $$f(x)=\frac{1}{x^{2}},[-1,2]$$

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