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Chapter 4: Problem 14

Differentiate the following functions. $$f(t)=\left(t^{3}-3 t\right) e^{1+t}$$

Short Answer

Expert verified
The derivative is \( f'(t) = e^{1+t} (t^3 + 3t^2 - 3t - 3) \).

Step by step solution

01

Identify the product rule components

The function given is a product of two functions: \[ f(t) = (t^3 - 3t) \times e^{1+t} \] Let \(u(t) = t^3 - 3t\) and \(v(t) = e^{1+t}\).
02

Differentiate \(u(t)\)

Find the derivative of \(u(t) = t^3 - 3t\): \[ u'(t) = 3t^2 - 3 \]
03

Differentiate \(v(t)\)

Find the derivative of \(v(t) = e^{1+t}\). Since \(e^{1+t}\) can be rewritten as \(e \times e^t\), using the chain rule: \[ v'(t) = e^{1+t} \times 1 = e^{1+t} \]
04

Apply the product rule

The product rule states \((uv)' = u'v + uv'\). Substitute \(u(t), u'(t), v(t), v'(t)\) into the product rule: \[ f'(t) = (3t^2 - 3)e^{1+t} + (t^3 - 3t)e^{1+t} \]
05

Simplify the expression

Factor out \(e^{1+t}\) from the two terms: \[ f'(t) = e^{1+t}[(3t^2 - 3) + (t^3 - 3t)] \]Combine like terms inside the brackets: \[ f'(t) = e^{1+t} (t^3 + 3t^2 - 3t - 3) \]

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Key Concepts

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

Product Rule
The product rule is a formula used to find the derivative of the product of two functions. Understanding when and how to apply this rule is essential for calculus.
It is given by \((uv)' = u'v + uv'\), where \(u\) and \(v\) are functions of \(t\), and \(u'\) and \(v'\) are their derivatives.
Applying this rule allows us to deal with more complex products of functions without simplifying them first.
Chain Rule
The chain rule is used to differentiate composite functions. It allows us to find the derivative of a function composed of two or more nested functions.
The chain rule states that if you have a function \(y = g(f(x))\), then the derivative \(dy/dx\) is given by\(dy/dx = g'(f(x)) \times f'(x)\).
This rule is particularly useful when dealing with functions like \(e^{1+t}\). We apply the chain rule by identifying the outer function and its inner function, differentiating each, and then multiplying the results.
Derivative
A derivative represents the rate at which a function is changing at any given point. It is an essential concept in calculus, showing how a function varies.
For example, to differentiate \(u(t) = t^3 - 3t\), we find the derivative to be \(u'(t) = 3t^2 - 3\).
Similarly, applying the chain rule to \(v(t) = e^{1+t}\), we get the derivative \(v'(t) = e^{1+t}\).
These individual derivatives are essential in applying the product rule to find the derivative of the given function \(f(t) = (t^3 - 3t) e^{1+t}\).

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