Problem 48 Differentiate each function. $... [FREE SOLUTION] (2024)

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Chapter 1: Problem 48

Differentiate each function. $$f(t)=\frac{3 t^{2}+2 t-1}{-t^{2}+4 t+1}$$

Short Answer

Expert verified

The derivative is \[ f'(t) = \frac{-6t^3 + 34t^2 + 12t + 6}{(-t^2 + 4t + 1)^2} \].

Step by step solution

- Identify the functions in the numerator and denominator

The function provided is $f(t) = \frac{3t^2 + 2t - 1}{-t^2 + 4t + 1}$. Here, the numerator is $u(t) = 3t^2 + 2t - 1$ and the denominator is $v(t) = -t^2 + 4t + 1$.

- Differentiate the numerator and denominator

Differentiate both the numerator and the denominator with respect to $t$.The derivative of the numerator: $u'(t) = \frac{d}{dt}(3t^2 + 2t - 1) = 6t + 2$.The derivative of the denominator: $v'(t) = \frac{d}{dt}(-t^2 + 4t + 1) = -2t + 4$.

- Apply the quotient rule

The quotient rule states that if $f(t) = \frac{u(t)}{v(t)}$, then $f'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2}$. Apply this rule using the expressions for $u(t)$, $v(t)$, $u'(t)$, and $v'(t)$ identified in Steps 1 and 2.Substitute the expressions to get: \[f'(t) = \frac{(6t + 2)(-t^2 + 4t + 1) - (3t^2 + 2t - 1)(-2t + 4)}{(-t^2 + 4t + 1)^2} \]

- Simplify the numerator of the derivative

Expand and simplify the numerator of the derivative calculated in Step 3. \[ (6t + 2)(-t^2 + 4t + 1) - (3t^2 + 2t - 1)(-2t + 4) \]Expanding the terms: \[ = -6t^3 + 24t^2 + 6t - 2t^2 + 8t + 2 - 6t^3 + 12t^2 - 2t + 4 \]Combine like terms to simplify: \[ = -6t^3 + 24t^2 + 6t - 2t^2 + 8t + 2 - 6t^3 + 12t^2 - 2t + 4 \]Simplify: \[ = -6t^3 + 34t^2 + 12t + 6 \]

- Simplify the denominator of the derivative

Simplify the denominator of the derivative, which is \[ (-t^2 + 4t + 1)^2 \].Thus, \[ f'(t) = \frac{-6t^3 + 34t^2 + 12t + 6}{(-t^2 + 4t + 1)^2} \].

Key Concepts

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

Calculus

Calculus is a branch of mathematics that studies how things change. It is divided into differential calculus and integral calculus. Differential calculus focuses on how functions change and how to measure that change, which is done through derivatives. Integral calculus, on the other hand, deals with summing up quantities and finding areas under curves. The fundamental theorem of calculus links these two branches, showing how differentiation and integration are interconnected. In this guide, we will primarily focus on differentiation.

Derivative

A derivative is a key concept in calculus that represents the rate at which a function changes at any given point. In simpler terms, it measures how the output of a function responds to changes in its input. The notation for the derivative of a function $ f(t) $ is $ f'(t) $. There are several rules for computing derivatives, such as the power rule, product rule, and quotient rule. For example, if $ f(t) = t^2 $, then its derivative $ f'(t) = 2t $, which means that for each unit increase in $ t $, $ f(t) $ increases by $ 2t $. Knowing how to find and interpret derivatives is essential in calculus.

Quotient Rule

The Quotient Rule is a method used in calculus for differentiating functions that are expressed as the quotient, or division, of two other functions. If you have a function $ f(t) = \frac{u(t)}{v(t)} $, where both $ u(t) $ and $ v(t) $ are differentiable, the Quotient Rule states that the derivative is:

\[ f'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2} \]

Let's break this down:

First differentiate the numerator $ u(t) $, which is $ u'(t) $.
Then differentiate the denominator $ v(t) $, which is $ v'(t) $.
Multiply $ u'(t) $ by $ v(t) $ and $ u(t) $ by $ v'(t) $.
Subtract the second product from the first.
Finally, divide the result by the square of the original denominator $ [v(t)]^2 $.

This rule simplifies the process of differentiating quotients, making it easier to handle complex functions like the one in the given exercise.

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Most popular questions from this chapter

For each of the following, graph $f$ and $f^{\prime}$ and then determine$f^{\prime}(1) .$ For Exercises use Deriv on the $T I-83$. $$f(x)=20 x^{3}-3 x^{5}$$Differentiate each function. $$F(t)=\frac{1}{t-4}$$Differentiate each function. $$g(x)=\left(x^{3}-8\right) \cdot \frac{x^{2}+1}{x^{2}-1}$$Graph each of the following. Then estimate the x-values at which tangent linesare horizontal. $$f(x)=10.2 x^{4}-6.9 x^{3}$$For each function, find the interval(s) for which $f^{\prime}(x)$ is positive. Find the points on the graph of $y=x^{4}-\frac{4}{3} x^{2}-4$ at which the tangent line is horizontal.

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Problem 48 Differentiate each function. $... [FREE SOLUTION] (2024)

Short Answer

Step by step solution

- Identify the functions in the numerator and denominator

- Differentiate the numerator and denominator

- Apply the quotient rule

- Simplify the numerator of the derivative

- Simplify the denominator of the derivative

Key Concepts

Calculus

Derivative

Quotient Rule

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Discrete Mathematics

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Privacy Overview

References