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Chapter 3: Problem 16

Find the indicated derivative. $$\frac{d}{d t}\left(\frac{2 t^{3}+1}{t^{4}}\right)$$

Short Answer

Expert verified
The derivative of \(\frac{2 t^{3}+1}{t^{4}}\) with respect to \(t\) is \(-2t^{-2} - 4t^{-5}\)

Step by step solution

01

Define u(t) and v(t)

Let \(u(t) = 2t^{3}+1\) and let \(v(t) = t^{4}\). Find their derivatives: \(u'(t) = 6t^{2}\) and \(v'(t) = 4t^{3}\)
02

Apply the Quotient Rule

By the Quotient Rule for derivatives, the derivative of \(u(t)/v(t)\) is \((v(t)u'(t) - u(t)v'(t))/v(t)^{2}\). Substitute the functions and their derivatives into this formula. This gives: \((t^{4} * 6t^{2} - (2t^{3}+1) * 4t^{3})/t^{8}\)
03

Simplify the Expression

Expand the terms in the numerator and simplify the entire expression. This results in: \((6t^{6} - 8t^{6} - 4t^{3})/t^{8}\). Combine like terms to get \((- 2t^{6} - 4t^{3})/t^{8}\). Divide each term in the numerator by \(t^{8}\) to get \(-2t^{-2} - 4t^{-5}\)

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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 continuous change, which has two fundamental operations: differentiation and integration. Differentiation focuses on understanding how a function's value changes as its input changes鈥攊t's about finding rates of change, commonly known as derivatives. In contrast, integration accumulates quantities, such as areas under curves.
Differentiation Techniques
Differentiation techniques are mathematical tools used to find derivatives. They range from basic methods for simple functions, like the power rule for monomials, to more complex rules for composite functions, such as the product rule, the quotient rule, and the chain rule.

Quotient Rule Overview

One crucial technique is the quotient rule, which is used to differentiate functions that are divided by each other, known as rational functions. It stipulates that the derivative of a function written as \(\frac{u(t)}{v(t)}\) is \(\frac{v(t)u'(t) - u(t)v'(t)}{[v(t)]^{2}}\), where \(u(t)\) is the numerator and \(v(t)\) is the denominator.
Derivatives of Polynomial Functions
Polynomial functions are expressions that consist of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and integer exponents. To differentiate polynomial functions, we often use standard rules like the power rule, which states that the derivative of \(ax^n\) is \(nax^{n-1}\).

Applying the Power Rule

In the case of the problem at hand, we differentiate \(2t^{3}\) using the power rule to obtain \(6t^{2}\) as the derivative. Such knowledge of deriving polynomial functions becomes a cornerstone when applying advanced techniques like the quotient rule.
Simplifying Expressions
Simplifying expressions in calculus makes equations easier to work with and interpret. After applying differentiation rules, we often combine like terms and factor expressions to simplify the result. In some cases, we divide each term by a common factor to reduce the expression to its simplest form.

Streamlining the Final Answer

For instance, in the provided example after applying the quotient rule, we simplify the numerator and cancel out terms with the denominator \(t^{8}\) to arrive at \( -2t^{-2} - 4t^{-5}\), which is a simplified form of the derivative we sought.

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Most popular questions from this chapter

Given two functions \(f\) and \(g,\) show that if \(f\) and \(f+g\) are differentiable, then \(y\) is differentiable. Give an example to show that the differentiable of \(f+g\) does not imply that \(f\) and \(g\) are each differentiable.

Find the numbers \(x\) for which (a) \(f^{\prime \prime}(x)=0\), (b) \(f^{\prime \prime}(x)>0,\) (c) \(f^{\prime \prime}(x)<0\). $$f(x)=x^{4}+2 x^{3}-12 x^{2}$$

Find the points \((c, f(c))\) where the line tangent to the graph of \(f(x)=x^{3}-x\) is parallel to the secant line that passes through the points \((-1, f(-1))\) and \((2, f(2))\).

Two curves are said to be orthogonal iff, at each point of intersection, the angle between them is a right angle. Show that the curves given are orthogonal. The ellipse \(3 x^{2}+2 y^{2}=5\) and \(y^{3}=x^{2}.\) HINT: The curves intersect at ( 1,1 ) and (-1,1)

Find a function \(y=f(x)\) with the given derivative. Check your answer by differentiation. $$y^{\prime}=3\left(x^{2}+1\right)^{2}(2 x)$$
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