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Chapter 3: Problem 9

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(q)=q^{3}+10$$

Short Answer

Expert verified
The derivative is \( 3q^2 \).

Step by step solution

01

Identify the Expression to Differentiate

The function given is \( f(q) = q^3 + 10 \). We need to find its derivative with respect to \( q \).
02

Differentiate Each Term Separately

Apply the power rule of differentiation to \( q^3 \), which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \). The derivative of \( 10 \), a constant, is \( 0 \).
03

Apply the Power Rule to \( q^3 \)

Using the power rule, the derivative of \( q^3 \) is \( 3q^{3-1} = 3q^2 \).
04

Combine the Derivatives

Combine the derivatives from the previous steps: the derivative of \( q^3 \) is \( 3q^2 \) and the derivative of the constant \( 10 \) is \( 0 \). Thus, the derivative of the entire function is \( 3q^2 + 0 \), which simplifies to \( 3q^2 \).

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Key Concepts

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

Power Rule
The power rule is a fundamental principle in the world of calculus. It's a straightforward and quick way to find the derivative of a function where the variable is raised to a power.
  • If you have a function of the form \( f(x) = x^n \), the derivative of this function is \( f'(x) = nx^{n-1} \).
  • This simply means you multiply the current power by the variable's coefficient (which is 1 if not written), and then decrease the power by one.
In our example, \( q^3 \), the power rule tells us to take the exponent 3, multiply it by the coefficient of the term (implicit 1 in this case), and subtract one from the power.This results in \( 3q^{3-1} \) or \( 3q^2 \). The power rule vastly simplifies the differentiation process and is easy to apply even in more complex functions.
Derivative of a Constant
The derivative of a constant is one of the most straightforward rules in calculus that makes working with functions easier.
  • A constant is a fixed value that does not change, such as 10 in our example.
  • The rule states that the derivative of any constant is zero. Why? Because a constant doesn't change, and therefore, its rate of change is zero.
In other words, when differentiating \( f(q) = q^3 + 10 \), the term 10 does not contribute to the derivative. We simply add zero for the constant part.This simplicity helps when handling complicated functions that include constant terms.
Calculus Basics
Calculus is a branch of mathematics that focuses on rates of change and accumulation. It's divided into two main areas: differential calculus and integral calculus.
  • Differential Calculus: Concentrates on the concept of the derivative, which represents the rate of change or the slope of a curve at any given point.
  • Integral Calculus: Primarily deals with accumulation of quantities and the areas under and between curves.
Understanding these basics can greatly help in studying many real-world phenomena like motion, growth, and trend analysis.Specifically, in differential calculus, finding a derivative like in our problem \( f(q)=q^3+10 \) showcases how calculus is used to determine dynamic changes in a function.Mastering these concepts gives you powerful tools to analyze and solve complex mathematical problems.

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Most popular questions from this chapter

A boat at anchor is bobbing up and down in the sea. The vertical distance, \(y,\) in feet, between the sea floor and the boat is given as a function of time, \(t,\) in minutes, by $$y=15+\sin (2 \pi t)$$. (a) Find the vertical velocity, \(v\), of the boat at time \(t\). (b) Make rough sketches of \(y\) and \(v\) against \(t\).

Find the equation of the tangent line to the graph of \(y=3^{x}\) at \(x=1 .\) Check your work by sketching a graph of the function and the tangent line on the same axes.

Differentiate the functions in Problems. Assume that \(A\) \(B,\) and \(C\) are constants. $$P(t)=12.41(0.94)^{t}$$

Let \(f(x)=x^{3}-6 x^{2}-15 x+20 .\) Find \(f^{\prime}(x)\) and all values of \(x\) for which \(f^{\prime}(x)=0 .\) Explain the relationship between these values of \(x\) and the graph of \(f(x)\)

Differentiate the functions in Problems \(1-20 .\) Assume that \(A\) and \(B\) are constants. $$z=\frac{e^{t^{2}}+t}{\sin (2 t)}$$
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