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Chapter 3: Problem 80

Use the properties of logarithms to simplify the following functions before computing \(f^{\prime}(x)\). $$f(x)=\log _{2} \frac{8}{\sqrt{x+1}}$$

Short Answer

Expert verified
Question: Find the derivative of the given logarithmic function: \(f(x)=\log _{2} \frac{8}{\sqrt{x+1}}\) Answer: The derivative of the given function is \(f^{\prime}(x)=-\frac{1}{2(x+1)\ln(2)}\).

Step by step solution

01

Simplify the given function using logarithm properties

We will first simplify the function using the properties of logarithms mentioned above. Given function: $$f(x)=\log _{2} \frac{8}{\sqrt{x+1}}$$ Using property 2, we can rewrite the function as: $$f(x)=\log _{2}(8) - \log _{2}(\sqrt{x+1})$$ Now, using property 3, we can rewrite the function as: $$f(x)=\log _{2}(2^3) - \frac{1}{2}\log_2(x+1)$$ Since \(\log_2(2^n)=n\), the simplified function is: $$f(x)=3 - \frac{1}{2}\log_2(x+1)$$
02

Compute the derivative of the simplified function

Now that we have the simplified function, we can compute its derivative \(f^{\prime}(x)\). First, let's calculate the derivative of \(\log_2(x+1)\): $$\frac{d}{dx}\log_2(x+1)=\frac{1}{(x+1)\ln(2)}$$ Now, calculate the derivative of \(3 - \frac{1}{2}\log_2(x+1)\) using the chain and power rule: $$f^{\prime}(x)=0 - \frac{1}{2} \cdot \frac{1}{(x+1)\ln(2)}$$ Simplifying, we get the derivative as: $$f^{\prime}(x)=-\frac{1}{2(x+1)\ln(2)}$$ So, the derivative of the given function is: $$f^{\prime}(x)=-\frac{1}{2(x+1)\ln(2)}$$

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Key Concepts

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

Logarithmic Differentiation
Logarithmic differentiation is a technique used in calculus to simplify the process of finding derivatives of certain complex functions. This method is particularly useful when dealing with functions involving products, quotients, or powers. By applying the properties of logarithms, we can transform the function into a form that makes differentiation easier.

Here are the basic steps:
  • Take the natural logarithm of both sides of the equation if it's not already expressed in logarithmic form.
  • Use properties of logarithms to simplify the expression.
  • Differentiate each side of the logarithmic equation with respect to the variable.
  • Solve for the derivative by substituting back any variables if necessary.
Logarithmic differentiation allows for simplification, especially when a function is a multiplication or division of multiple parts. In our exercise, we used the properties of logarithms to simplify the division inside the log function. This made taking the derivative more straightforward.
Derivatives
Derivatives are fundamental in calculus. They help us understand how a function changes at any given point. The derivative of a function reflects the rate of change or the slope of a function at a particular point.

Some key concepts to remember about derivatives:
  • The power rule is useful for functions of the form \(x^n\). The derivative is \(nx^{n-1}\).
  • Chain rule applies when differentiating composites of functions, such as \(g(f(x))\), and it states that the derivative is \(g'(f(x)) \cdot f'(x)\).
  • For logarithmic functions, the derivative of \(\log_b(x)\) is \( \frac{1}{x\ln(b)} \).
In our solved exercise, we used these principles by first simplifying the expression to make the derivative calculation easier, expressing the problem in terms of a base-2 logarithm, and applying these derivative rules.
Properties of Logarithms
The properties of logarithms play a crucial role in simplifying expressions and solving calculus problems. There are three main properties you should be familiar with:

  • Product property: \(\log_b(MN) = \log_b(M) + \log_b(N)\).
  • Quotient property: \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\).
  • Power property: \(\log_b(M^n) = n\log_b(M)\).
In our exercise, we applied these properties to break down a complex logarithmic expression into simpler parts. The original function was a logarithm dividing two expressions, which was refined into a subtraction through the quotient property. Additionally, the power property was used to handle the square root term, simplifying the problem for easier differentiation. Understanding and applying these properties is indispensable in solving logarithm-involved calculus problems efficiently.

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Most popular questions from this chapter

Recall that \(f\) is even if \(f(-x)=f(x),\) for all \(x\) in the domain of \(f,\) and \(f\) is odd if \(f(-x)=-f(x),\) for all \(x\) in the domain of \(f\). a. If \(f\) is a differentiable, even function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither. b. If \(f\) is a differentiable, odd function on its domain, determine whether \(f^{\prime}\) is even, odd, or neither.

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 \(h=40\)

Earth's atmospheric pressure decreases with altitude from a sea level pressure of 1000 millibars (a unit of pressure used by meteorologists). Letting \(z\) be the height above Earth's surface (sea level) in \(\mathrm{km}\), the atmospheric pressure is modeled by \(p(z)=1000 e^{-z / 10}.\) a. Compute the pressure at the summit of Mt. Everest which has an elevation of roughly \(10 \mathrm{km}\). Compare the pressure on Mt. Everest to the pressure at sea level. b. Compute the average change in pressure in the first \(5 \mathrm{km}\) above Earth's surface. c. Compute the rate of change of the pressure at an elevation of \(5 \mathrm{km}\). d. Does \(p^{\prime}(z)\) increase or decrease with \(z\) ? Explain. e. What is the meaning of \(\lim _{z \rightarrow \infty} p(z)=0 ?\)

Prove the following identities and give the values of \(x\) for which they are true. $$\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}$$

A cylindrical tank is full at time \(t=0\) when a valve in the bottom of the tank is opened. By Torricelli's Law, the volume of water in the tank after \(t\) hours is \(V=100(200-t)^{2}\), measured in cubic meters. a. Graph the volume function. What is the volume of water in the tank before the valve is opened? b. How long does it take the tank to empty? c. Find the rate at which water flows from the tank and plot the flow rate function. d. At what time is the magnitude of the flow rate a minimum? A maximum?
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