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Chapter 3: Problem 13

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$f(x)=5 x^{3}$$

Short Answer

Expert verified
Answer: The derivative of the function \(f(x) = 5x^3\) is \(f'(x) = 15x^2\).

Step by step solution

01

Identify the function and its components

The given function is: $$f(x) = 5x^3$$ Here, the value of \(a=5\) and \(n=3\).
02

Apply the power rule

Now, we will apply the power rule to the function \(f(x) = 5x^3\). For the function \(ax^n\), the derivative is given by: $$\frac{d}{dx}(ax^n) = n(ax^{n-1})$$ So, for our function: $$\frac{d}{dx}(5x^3) = 3(5x^{3-1})$$
03

Simplify the derivative

Next, we simplify the derivative as follows: $$f'(x) = 3(5x^{2})$$ $$f'(x) = 15x^2$$ The derivative of the function \(f(x) = 5x^3\) is: $$f'(x) = 15x^2$$

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Key Concepts

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

Power Rule
Understanding the power rule is essential when taking the derivative of functions in calculus, especially when dealing with polynomial terms like those found in the function f(x) = 5x^3. The power rule is a straightforward but mighty tool, stating that if you have a function of the form ax^n, where a is a constant and x is raised to an exponent n, the derivative of this function with respect to x is anx^{(n-1)}.

When applying the power rule, you are essentially reducing the exponent by one and multiplying by the original exponent. This method streamlines the process of finding the derivative and can be quickly applied to each term individually when working with polynomials. It's important to remember that the power rule can only be used when the exponent is a real number, which makes it especially suited for functions like the one in our example.
Simplifying Derivatives
Simplifying derivatives is a vital step in calculus as it helps to make complex expressions more manageable and easier to work with. After applying the power rule to our function f(x) = 5x^3, we get the derivative f'(x) = 3(5x^2). To simplify this expression, we follow basic arithmetic and algebra rules to combine like terms and constants.

For this derivative, we multiply the constant outside the parenthesis by each term inside. In this case, 3 multiplied by 5 results in 15, and the x^2 remains unchanged. Thus, the significantly simpler and final form of the derivative is f'(x) = 15x^2. Simplification is critical for making further calculations easier, such as evaluating the derivative at specific points or integrating the function.
Calculus
Calculus, the branch of mathematics that studies the changes between values, is perfectly exemplified by the concept of derivatives. The derivative is a tool that allows us to determine the instantaneous rate of change of a function, or, in simpler terms, how a function is changing at any given point along its curve.

The derivative of a function at a point is the slope of the tangent line to the function's graph at that point. This is analogous to finding the speed of an object at a specific moment in time in a physical context. In the example with f(x) = 5x^3, calculating the derivative gives us a formula f'(x) = 15x^2 which can predict how quickly the values of f(x) are changing in relation to changes in x. This fundamental concept is not only foundational in mathematics but also across many scientific disciplines including physics, engineering, economics, and beyond, showcasing the universal importance of calculus.

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Most popular questions from this chapter

Logistic growth Scientists often use the logistic growth function \(P(t)=\frac{P_{0} K}{P_{0}+\left(K-P_{0}\right) e^{-r_{d}}}\) to model population growth, where \(P_{0}\) is the initial population at time \(t=0, K\) is the carrying capacity, and \(r_{0}\) is the base growth rate. The carrying capacity is a theoretical upper bound on the total population that the surrounding environment can support. The figure shows the sigmoid (S-shaped) curve associated with a typical logistic model. World population (part 1 ) The population of the world reached 6 billion in \(1999(t=0)\). Assume Earth's carrying capacity is 15 billion and the base growth rate is \(r_{0}=0.025\) per year. a. Write a logistic growth function for the world's population (in billions) and graph your equation on the interval \(0 \leq t \leq 200\) using a graphing utility. b. What will the population be in the year 2020? When will it reach 12 billion?

Vertical tangent lines a. Determine the points where the curve \(x+y^{2}-y=1\) has a vertical tangent line (see Exercise 53 ). b. Does the curve have any horizontal tangent lines? Explain.

Gone fishing When a reservoir is created by a new dam, 50 fish are introduced into the reservoir, which has an estimated carrying capacity of 8000 fish. A logistic model of the fish population is \(P(t)=\frac{400,000}{50+7950 e^{-0.5 t}},\) where \(t\) is measured in years.c. How fast (in fish per year) is the population growing at \(t=0 ?\) At \(t=5 ?\) d. Graph \(P^{\prime}\) and use the graph to estimate the year in which the population is growing fastest.

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas \(y=c x^{2}\) form orthogonal trajectories with the family of ellipses \(x^{2}+2 y^{2}=k,\) where \(c\) and \(k\) are constants (see figure). Find \(d y / d x\) for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. \(y=m x ; x^{2}+y^{2}=a^{2},\) where \(m\) and \(a\) are constants

The total energy in megawatt-hr (MWh) used by a town is given by $$E(t)=400 t+\frac{2400}{\pi} \sin \frac{\pi t}{12},$$ where \(t \geq 0\) is measured in hours, with \(t=0\) corresponding to noon. a. Find the power, or rate of energy consumption, \(P(t)=E^{\prime}(t)\) in units of megawatts (MW). b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day? c. At what time of day is the rate of energy consumption a minimum? What is the power at that time of day? d. Sketch a graph of the power function reflecting the times at which energy use is a minimum or maximum.
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