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Chapter 3: Problem 41

If \(\frac{d}{d t}(t f(t))=1+f(t),\) what is \(f^{\prime}(t) ?\)

Short Answer

Expert verified
The derivative of the function is \( f'(t) = \frac{1}{t} \).

Step by step solution

01

Differentiate the Product

We have an equation: \( \frac{d}{dt}(t f(t)) = 1 + f(t) \). Start by using the product rule to differentiate \( t f(t) \), which gives: \( f(t) + t f'(t) \). Set this equal to the right side of the equation: \( 1 + f(t) \). This gives us the equation: \( f(t) + t f'(t) = 1 + f(t) \).
02

Simplify the Equation

From the equation \( f(t) + t f'(t) = 1 + f(t) \), subtract \( f(t) \) from both sides to isolate terms involving \( f'(t) \). This simplifies to: \( t f'(t) = 1 \).
03

Solve for the Derivative

Now that we have the simplified equation \( t f'(t) = 1 \), divide both sides by \( t \) to solve for \( f'(t) \). This yields \( f'(t) = \frac{1}{t} \).

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Key Concepts

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

Product Rule Differentiation
When dealing with the differentiation of a product of two functions, the product rule comes in handy. The product rule is a formula used to find the derivative of a product of two functions. If you have a function like \( h(t) = u(t) \cdot v(t) \), the product rule states that the derivative is given by: \[ h'(t) = u'(t) v(t) + u(t) v'(t) \] This means you differentiate the first function \( u(t) \) and multiply it by the second \( v(t) \), then add the product of the first function and the derivative of the second function.
  • This rule is crucial when you're working with two functions that multiply together, such as our case with \( t f(t) \).
  • In applying the product rule, it's essential to correctly identify which factor of the product you are differentiating with respect to and ensure accurate calculation for each part.
Grasping this principle helps simplify and solve more complex differential equations.
Derivatives
A derivative, in simple terms, represents the rate of change or the slope of a function at any point. Mathematically, if you have a function \( f(t) \), its derivative \( f'(t) \) provides information about how \( f(t) \) changes as \( t \) changes. Here are some key takeaways about derivatives:
  • They are fundamental in calculus, forming the basis for finding rates of change and slopes of curves.
  • They apply to various real-world scenarios, including motion, growth rates, and trends.
  • For our example, knowing \( f'(t) = \frac{1}{t} \) from \( t\cdot f'(t) = 1 \), tells us how \( f(t) \) changes with \( t \).
Understanding derivatives is crucial as they are foundational to more advanced calculus concepts and practical applications.
Solving Differential Equations
A differential equation involves an unknown function and its derivatives. Solving differential equations means finding a function that satisfies a given relationship involving its derivatives. In this context:
  • Our task was to solve \( \frac{d}{d t}(t f(t))=1+f(t) \) for \( f'(t) \).
  • By using the product rule, simplifying, and isolating \( f'(t) \), we find \( f'(t) = \frac{1}{t} \).
In general, the process involves several steps:
  • Begin by simplifying and rewriting the equation based on derivatives' properties.
  • Manipulate the equation algebraically to isolate the desired variable or function.
  • This often involves integrating or differentiating both sides of an equation.
Mastering how to solve differential equations empowers you to tackle a wide array of problems in physics, engineering, and beyond.

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Most popular questions from this chapter

If the demand curve is a line, we can write \(p=b+m q\) where \(p\) is the price of the product, \(q\) is the quantity sold at that price, and \(b\) and \(m\) are constants. (a) Write the revenue as a function of quantity sold. (b) Find the marginal revenue function.

Differentiate the functions in Problems. Assume that \(A\) \(B,\) and \(C\) are constants. $$f(t)=e^{3 t}$$

Find the quadratic polynomial \(g(x)=a x^{2}+b x+c\) which best fits the function \(f(x)=e^{x}\) at \(x=0,\) in the sense that $$g(0)=f(0), \text { and } g^{\prime}(0)=f^{\prime}(0), \text { and } g^{\prime \prime}(0)=f^{\prime \prime}(0)$$ Using a computer or calculator, sketch graphs of \(f\) and \(g\) on the same axes. What do you notice?

The value of an automobile purchased in 2009 can be approximated by the function \(V(t)=25(0.85)^{t},\) where \(t\) is the time, in years, from the date of purchase, and \(V(t)\) is the value, in thousands of dollars. (a) Evaluate and interpret \(V(4),\) including units. (b) Find an expression for \(V^{\prime}(t),\) including units. (c) Evaluate and interpret \(V^{\prime}(4),\) including units. (d) Use \(V(t), V^{\prime}(t),\) and any other considerations you think are relevant to write a paragraph in support of or in opposition to the following statement: "From a monetary point of view, it is best to keep this vehicle as long as possible."

The quantity demanded of a certain product, \(q\), is given in terms of \(p,\) the price, by $$q=1000 e^{-0.02 p}$$ (a) Write revenue, \(R,\) as a function of price. (b) Find the rate of change of revenue with respect to price. (c) Find the revenue and rate of change of revenue with respect to price when the price is \(\$ 10 .\) Interpret your answers in economic terms.
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