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Chapter 3: Problem 32

Find the derivative. Assume \(a, b, c, k\) are constants. $$f(x)=k x^{2}$$

Short Answer

Expert verified
The derivative is \(f'(x) = 2kx\).

Step by step solution

01

Identify the Derivative Formula

The function to be differentiated is a power function of the form \(f(x)=k x^{2}\). The derivative of a power function \(kx^n\) is given by \(f'(x) = nkx^{n-1}\).
02

Apply the Derivative Formula

Using the formula derived in Step 1, apply it to find the derivative \(f'(x) = 2k x^{2-1} = 2k x^1\). This simplifies to \(f'(x) = 2kx\).

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Key Concepts

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

Derivative
A derivative is a fundamental concept in calculus, representing the rate of change of a function with respect to a variable. In simpler words, it tells us how fast or slow a function's value is changing at any given point. For instance, if you think of a function as describing the position of a car over time, the derivative tells you the car's speed.
  • The notation for a derivative is often written as \(f'(x)\) or \(\frac{df}{dx}\).
  • The derivative provides a linear approximation of how the function behaves locally near a point.
  • In physical terms, the derivative can be related to real-world rates, like velocity, growth rates, or slopes of curves.
Calculating derivatives involves using specific rules and formulas, making the process systematic and straightforward once you understand the concepts. With practice, identifying and applying these rules becomes more intuitive.
Power Function
A power function is a type of mathematical function of the form \(f(x) = kx^n\), where \(k\) is a constant coefficient, and \(n\) is a non-zero real number. Both \(k\) and \(n\) determine the shape and position of the function graph on a coordinate plane.
When differentiating power functions, it helps to understand their general characteristics:
  • Power functions can be simple linear functions if \(n = 1\), or they can be more complex quadratic, cubic, etc., if \(n > 1\).
  • The constant \(k\) scales the function, affecting its steepness and direction.
  • Power functions are smooth and continuous, making them predictable and easy to model in various scenarios.
This general formula \(kx^n\) simplifies the process of differentiation by helping us quickly determine how these functions will behave when their derivatives are taken.
Differentiation Formula
The differentiation formula for power functions allows us to find the rate of change efficiently. For a power function \(f(x) = kx^n\), the formula for the derivative is given by \(f'(x) = nkx^{n-1}\). This rule, known as the power rule, is among the most essential tools in calculus.
Let's break down the power rule:
  • Multiply the exponent \(n\) by the coefficient \(k\).
  • Subtract one from the exponent \(n\) to adjust the degree of the power function.
  • Apply these operations systematically to find the derivative.
Using the earlier example of \(f(x) = kx^2\), the application gives us \(f'(x) = 2kx^{2-1} = 2kx\), offering a clear method of computing derivatives efficiently without needing a new model for every problem. Understanding and applying these formulas can clarify various changes and trends in both academic and real-life situations.

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Most popular questions from this chapter

If you invest \(P\) dollars in a bank account at an annual interest rate of \(r \%,\) then after \(t\) years you will have \(B\) dollars, where $$B=P\left(1+\frac{r}{100}\right)^{t}$$ (a) Find \(d B / d t,\) assuming \(P\) and \(r\) are constant. In terms of money, what does \(d B / d t\) represent? (b) Find \(\overline{d B} / d r,\) assuming \(P\) and \(t\) are constant. In terms of money, what does \(d B / d r\) represent?

Differentiate the functions in Problems. Assume that \(A\) \(B,\) and \(C\) are constants. $$f(t)=A e^{t}+B \ln t$$

The cost to produce \(q\) items is \(C(q)=1000+2 q^{2}\) dollars. Find the marginal cost of producing the \(25^{\text {th }}\) item. Interpret your answer in terms of costs.

The quantity of a drug, \(Q\) mg, present in the body \(t\) hours after an injection of the drug is given is $$Q=f(t)=100 t e^{-0.5 t}$$ Find \(f(1), f^{\prime}(1), f(5),\) and \(f^{\prime}(5) .\) Give units and interpret the answers.

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.
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