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Chapter 3: Problem 46

If \(f(t)=t^{4}-3 t^{2}+5 t\), find \(f^{\prime}(t)\) and \(f^{\prime \prime}(t)\).

Short Answer

Expert verified
The first derivative is \( f'(t) = 4t^3 - 6t + 5 \) and the second derivative is \( f''(t) = 12t^2 - 6 \).

Step by step solution

01

Apply Power Rule to Find the First Derivative

The function given is \( f(t) = t^4 - 3t^2 + 5t \). To find the first derivative \( f'(t) \), we'll use the power rule: if \( f(t) = at^n \), then \( f'(t) = nat^{n-1} \). 1. For the term \( t^4 \), the derivative is \( 4t^3 \).2. For \( -3t^2 \), the derivative is \( -6t \).3. For \( 5t \), the derivative is \( 5 \). Thus, \( f'(t) = 4t^3 - 6t + 5 \).
02

Apply Power Rule to Find the Second Derivative

To find the second derivative \( f''(t) \), we differentiate the first derivative \( f'(t) = 4t^3 - 6t + 5 \).1. For the term \( 4t^3 \), apply the power rule to get \( 12t^2 \).2. For \( -6t \), the derivative is \( -6 \).3. The constant \( 5 \) has a derivative of \( 0 \).So, \( f''(t) = 12t^2 - 6 \).

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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 tool in calculus differentiation that simplifies finding the derivative of a power function. When you have a function in the form of \( f(t) = at^n \), the power rule allows you to find its derivative efficiently. The rule states:
  • Multiply the exponent \( n \) by the coefficient \( a \).
  • Subtract one from the exponent.

For instance, applying the power rule to \( t^4 \) involves multiplying \( 4 \) (the exponent) by \( 1 \) (the coefficient), resulting in \( 4t^3 \). This makes it easier to determine the rate at which functions change, which is essential for understanding behaviors like velocity and growth.
First Derivative
The first derivative of a function, often denoted as \( f'(t) \), provides crucial insight into the rate of change of the function. In simple terms, it tells us how fast or slow the function's value is changing at any given point. Using the power rule, let's break down the function \( f(t) = t^4 - 3t^2 + 5t \):
  • The first term \( t^4 \) becomes \( 4t^3 \) after differentiation.
  • \(-3t^2\) simplifies to \(-6t\).
  • The linear term \(5t\) reduces to the constant \(5\).

Thus, the first derivative, \( f'(t) = 4t^3 - 6t + 5 \), reflects the changes in slopes or the steepness of the original function at different points.
Second Derivative
The second derivative, \( f''(t) \), is essentially the derivative of the first derivative. This means we apply the differentiation process to \( f'(t) \). The second derivative provides insights into the concavity of the function, helping us determine if the function is curving upwards or downwards. Let's look at the steps for our given function's first derivative \( f'(t) = 4t^3 - 6t + 5 \):
  • The term \( 4t^3 \) becomes \( 12t^2 \) when differentiated again.
  • The term \(-6t\) results in \(-6\).
  • The constant \(5\) becomes \(0\), as constants do not change.

Therefore, the second derivative is \( f''(t) = 12t^2 - 6 \). This provides valuable information about the acceleration or deceleration of the rate of change itself.

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Most popular questions from this chapter

Find the derivative. Assume that \(a, b, c\), and \(k\) are constants. $$ y=5 x e^{x^{2}} $$

If \(p\) is price in dollars and \(q\) is quantity, demand for a product is given by $$ q=5000 e^{-0.08 p} $$ (a) What quantity is sold at a price of \(\$ 10\) ? (b) Find the derivative of demand with respect to price when the price is \(\$ 10\) and interpret your answer in terms of demand.

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