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Chapter 3: Problem 17

Find the derivative of the following functions. $$y=\ln x^{2}$$

Short Answer

Expert verified
Answer: The derivative of the function is \(y' = \frac{2}{x}\).

Step by step solution

01

Rewrite the function using properties of logarithms

Recall that the logarithm of a power can be rewritten as a product of the power and the logarithm of the base. So, we can rewrite the given function as: $$y = 2\ln x$$
02

Differentiate the rewritten function using the chain rule

To differentiate the given function, we will use the chain rule. Recall that the chain rule states that: $$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$ Now, applying the chain rule to our rewritten function, we differentiate \(\ln x\) with respect to \(x\) and multiply it by the derivative of \(2\): $$y' = 2\cdot\frac{1}{x}$$
03

Simplify the result

Now we need to simplify our result. We do this by multiplying 2 by the fraction: $$y' = \frac{2}{x}$$ The derivative of the given function, \(y = \ln x^2\), is \(y' = \frac{2}{x}\).

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Key Concepts

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

Logarithmic Functions
Logarithmic functions are mathematical expressions involving logarithms. One key property of logarithms is their ability to transform multiplication into addition and powers into products. For instance, when you have a power inside a logarithm, like \(x^2\), you can use the identity \(\ln(x^2) = 2\ln(x)\). This property is enormously helpful in calculus, as it simplifies complex expressions, thus making differentiation more straightforward. Remembering these properties will make working with logarithmic functions much easier.
  • Key Point 1: The logarithm of a power can be rewritten as the power times the logarithm of the base.
  • Key Point 2: This property helps in simplifying the differentiation process.
Chain Rule
The chain rule is a fundamental concept in calculus, essential for finding the derivative of composite functions. It allows us to differentiate a function by taking the derivative of the outer function and also the inner function, and then multiplying them. The formula for the chain rule is \((f(g(x)))' = f'(g(x)) \cdot g'(x)\).
In our exercise, we apply the chain rule by treating \(\ln(x)\) as our inner function and the multiplication by 2 as the outer function. By applying the chain rule, we are able to systematically find the derivative.
  • Key Point 1: Identify the inner and outer functions correctly to apply the chain rule.
  • Key Point 2: It's important to multiply the derivatives of both the inner and outer functions.
Differentiation
Differentiation is a key operation in calculus, used to find the rate at which a function is changing at any given point. When differentiating logarithmic functions, it's important to remember especially how to handle the \(\ln(x)\) function. The derivative of \(\ln(x)\) with respect to \(x\) is \(\frac{1}{x}\). In the exercise, after simplifying the logarithmic expression using properties, we directly differentiate using this result. The final step in differentiation is simplifying your expression, as seen when we got \(y' = \frac{2}{x}\) after differentiating \(2\ln(x)\).
  • Key Point 1: The derivative of \(\ln(x)\) is \(\frac{1}{x}\).
  • Key Point 2: Always simplify your result to get the cleanest form of the derivative.

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Most popular questions from this chapter

a. Derive a formula for the second derivative, \(\frac{d^{2}}{d x^{2}}(f(g(x)))\) b. Use the formula in part (a) to calculate \(\frac{d^{2}}{d x^{2}}\left(\sin \left(3 x^{4}+5 x^{2}+2\right)\right)\)

The lateral surface area of a cone of radius \(r\) and height \(h\) (the surface area excluding the base) is \(A=\pi r \sqrt{r^{2}+h^{2}}.\) a. Find \(d r / d h\) for a cone with a lateral surface area of \(A=1500 \pi.\) b. Evaluate this derivative when \(r=30\) and \(h=40.\)

Find \(d^{2} y / d x^{2}.\) $$2 x^{2}+y^{2}=4$$

Complete the following steps. a. Find equations of all lines tangent to the curve at the given value of \(x.\) b. Graph the tangent lines on the given graph. \(4 x^{3}=y^{2}(4-x) ; x=2\) (cissoid of Diocles) (Graph cant copy)

Suppose you forgot the Quotient Rule for calculating \(\frac{d}{d x}\left(\frac{f(x)}{g(x)}\right) .\) Use the Chain Rule and Product Rule with the identity \(\frac{f(x)}{g(x)}=f(x)(g(x))^{-1}\) to derive the Quotient Rule.
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