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Chapter 3: Problem 2

In Exercises \(1-18,\) find \(d y / d x\) $$ y=\frac{3}{x}+5 \sin x $$

Short Answer

Expert verified
\( \frac{dy}{dx} = -\frac{3}{x^2} + 5 \cos x \)

Step by step solution

01

Identify the components of the function

The given function is \( y = \frac{3}{x} + 5 \sin x \). It consists of two main terms: \( \frac{3}{x} \) and \( 5 \sin x \). We need to differentiate each term with respect to \( x \).
02

Differentiate the first term

The first term is \( \frac{3}{x} \), which can be rewritten as \( 3x^{-1} \). Using the power rule \( \frac{d}{dx} x^n = nx^{n-1} \), the derivative of \( 3x^{-1} \) is \( 3 \times (-1)x^{-2} = -\frac{3}{x^2} \).
03

Differentiate the second term

The second term is \( 5 \sin x \). The derivative of \( \sin x \) is \( \cos x \). Hence, using the constant multiple rule, the derivative of \( 5 \sin x \) is \( 5 \cos x \).
04

Combine the derivatives

Add the derivatives of each term together to find \( \frac{dy}{dx} \). Thus, \( \frac{dy}{dx} = -\frac{3}{x^2} + 5 \cos x \).

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Key Concepts

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

Calculus Exercises
Calculus exercises often involve finding derivatives, which are a fundamental aspect of differential calculus. In this exercise, the task is to find the derivative of a given function with respect to a specific variable, in this case, \( x \).
The given function is \( y = \frac{3}{x} + 5 \sin x \). This is a typical problem where multiple differentiation rules might be needed to solve it.
  • You encounter two main mathematical operations:
    • The rational function \( \frac{3}{x} \)
    • The trigonometric function \( 5 \sin x \)
By identifying these components, you set a strong foundation for applying specific rules to find the derivative of each part. Remember, practice with varied exercises helps in understanding the use and application of different derivative rules.
Power Rule Differentiation
The power rule is an essential technique in calculus for finding derivatives of polynomial expressions. It states that if you have \( x^n \), its derivative is \( nx^{n-1} \).
In our exercise, the term \( \frac{3}{x} \) is reframed using the power rule. This expression can be rewritten as \( 3x^{-1} \), a format suitable for applying the power rule.
  • Applying the power rule, differentiate \( 3x^{-1} \):
    • Bring down the exponent to multiply with the coefficient: \( 3 imes (-1) \)
    • Decrease the exponent by one: the new exponent is \( -2 \)
  • The resulting derivative is \( -\frac{3}{x^2} \). As seen, power rule differentiation simplifies the process of finding the derivative of functions with rational expressions.
Trigonometric Functions Derivatives
Trigonometric functions are common in calculus problems, and knowing their derivatives is crucial. In the given exercise, we have the function \( 5 \sin x \), where the derivative of \( \sin x \) is \( \cos x \).
Using the constant multiple rule with trigonometric derivatives, we differentiate \( 5 \sin x \):
  • Since the derivative of \( \sin x \) is \( \cos x \), multiply this derivative by the constant 5.
  • The derivative is thus \( 5 \cos x \).
Combining this with the derivative of the rational term gives us the complete derivative of the function. Understanding trigonometric derivatives expands your ability to solve a wider range of calculus problems.

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Most popular questions from this chapter

Constant acceleration Suppose that the velocity of a falling body is \(v=k \sqrt{s} \mathrm{m} / \mathrm{sec}(k\) a constant) at the instant the body has fallen \(s \mathrm{m}\) from its starting point. Show that the body's acceleration is constant.

Use a CAS to perform the following steps. \begin{equation} \begin{array}{l}{\text { a. Plot the equation with the implicit plotter of a CAS. Check to }} \\ {\text { see that the given point } P \text { satisfies the equation. }} \\ {\text { b. Using implicit differentiation, find a formula for the deriva- }} \\ {\text { tive } d y / d x \text { and evaluate it at the given point } P .}\end{array} \end{equation} \begin{equation} \begin{array}{l}{\text { c. Use the slope found in part (b) to find an equation for the tan- }} \\ {\text { gent line to the curve at } P \text { . Then plot the implicit curve and }} \\ {\text { tangent line together on a single graph. }}\end{array} \end{equation} \begin{equation} x \sqrt{1+2 y}+y=x^{2}, \quad P(1,0) \end{equation}

Estimate the allowable percentage error in measuring the diameter \(D\) of a sphere if the volume is to be calculated correctly to within 3\(\% .\)

Surface area Suppose that the radius \(r\) and surface area \(S=4 \pi r^{2}\) of a sphere are differentiable functions of \(t\) . Write an equation that relates \(d S / d t\) to \(d r / d t\) .

Estimating height of a building A surveyor, standing 30 \(\mathrm{ft}\) from the base of a building, measures the angle of elevation to the top of the building to be \(75^{\circ} .\) How accurately must the angle be measured for the percentage error in estimating the height of the building to be less than 4\(\% ?\)
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