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Chapter 3: Problem 3

In Exercises \(1-18,\) find \(d y / d x\) $$ y=x^{2} \cos x $$

Short Answer

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

Step by step solution

01

Identify the functions to differentiate

The function given is a product of two functions: \( y = x^2 \cdot \cos x \). This includes a polynomial term \( x^2 \) and a trigonometric term \( \cos x \). To differentiate \( y \) with respect to \( x \), we need to use the product rule.
02

State the product rule formula

The product rule formula is given by \( \frac{d}{dx}[u v] = u \frac{dv}{dx} + v \frac{du}{dx} \), where \( u = x^2 \) and \( v = \cos x \).
03

Differentiate each component

Now we differentiate each component separately. For \( u = x^2 \), the derivative \( \frac{du}{dx} = 2x \). For \( v = \cos x \), the derivative \( \frac{dv}{dx} = -\sin x \).
04

Apply the product rule

Substitute \( u \), \( \frac{du}{dx} \), \( v \), and \( \frac{dv}{dx} \) into the product rule formula: \[ \frac{d}{dx}(x^2 \cos x) = x^2 (-\sin x) + \cos x (2x) \]
05

Simplify the expression

Simplify the expression obtained from the product rule: \[ -x^2 \sin x + 2x \cos x \]Thus, \( \frac{dy}{dx} = -x^2 \sin x + 2x \cos x \).

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Key Concepts

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

Product Rule in Differentiation
When we differentiate a product of two functions, we use the product rule to find the derivative correctly.
  • The product rule helps us break down the differentiation of two multiplied functions.
  • The formula for the product rule is: \[ \frac{d}{dx}[uv] = u \frac{dv}{dx} + v \frac{du}{dx} \]
  • In simpler terms, differentiate the first function then multiply it by the second function, and add the first function multiplied by the derivative of the second function.
This step-by-step process ensures you capture the full behavior of the product under differentiation. It's important to properly identify each function in the product and differentiate them separately before applying the rule.
Understanding Trigonometric Functions in Calculus
Trigonometric functions like \( \cos x \) and \( \sin x \) are essential in calculus, especially when differentiating expressions that involve angles.
  • These functions often describe wave-like phenomena, such as sound or light.
  • In our original exercise, we differentiate \( \cos x \), resulting in \( -\sin x \).
  • This stems from standard differentiation rules for trigonometric functions.
Each trigonometric function has specific derivatives that are crucial to memorize for calculus. These derivatives tell us how the slope of the wave changes at any point.
Basics of Derivatives
Derivatives measure how a function changes as its input changes, and they're foundational in calculus for understanding slopes and rates of change.
  • The derivative of a function \( y = f(x) \) is often written as \( \frac{dy}{dx} \) or \( f'(x) \).
  • It gives us the slope of the tangent line to the function at any given point.
  • Derivatives are crucial in finding rates of change, optimization problems, and understanding the behavior of functions.
Gaining proficiency in calculating derivatives allows you to tackle a wide range of mathematical and real-world problems, making it a fundamental skill in mathematics.

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Most popular questions from this chapter

Find both \(d y / d x\) (treating \(y\) as a differentiable function of \(x\) ) and \(d x / d y\) (treating \(x\) as a differentiable function of \(y )\) . How do \(d y / d x\) and \(d x / d y\) seem to be related? Explain the relationship geometrically in terms of the graphs. \begin{equation} x y^{3}+x^{2} y=6 \end{equation}

Diagonals If \(x, y,\) and \(z\) are lengths of the edges of a rectangular box, the common length of the box's diagonals is \(s=\) \(\sqrt{x^{2}+y^{2}+z^{2}}\) a. Assuming that \(x, y,\) and \(z\) are differentiable functions of \(t\) how is \(d s / d t\) related to \(d x / d t, d y / d t,\) and \(d z / d t\) ? b. How is \(d s / d t\) related to \(d y / d t\) and \(d z / d t\) if \(x\) is constant? c. How are \(d x / d t, d y / d t,\) and \(d z / d t\) related if \(s\) is constant?

Find \(d y / d x\) if \(y=x^{3 / 2}\) by using the Chain Rule with \(y\) as a composite of $$\begin{array}{l}{\text { a. } y=u^{3} \text { and } u=\sqrt{x}} \\ {\text { b. } y=\sqrt{u} \text { and } u=x^{3}}\end{array}$$

Use a CAS to perform the following steps for the functions \begin{equation} \begin{array}{l}{\text { a. Plot } y=f(x) \text { to see that function's global behavior. }} \\ {\text { b. Define the difference quotient } q \text { at a general point } x, \text { with }} \\ {\text { general step size } h .} \\\ {\text { c. Take the limit as } h \rightarrow 0 . \text { What formula does this give? }} \\ {\text { d. Substitute the value } x=x_{0} \text { and plot the function } y=f(x)} \\ \quad {\text { together with its tangent line at that point. }} \\ {\text { e. Substitute various values for } x \text { larger and smaller than } x_{0} \text { into }} \\ \quad {\text { the formula obtained in part (c). Do the numbers make sense }} \\ \quad {\text { with your picture? }} \\ {\text { f. Graph the formula obtained in part (c). What does it mean }} \\ \quad {\text { when its values are negative? Zero? Positive? Does this make }} \\ \quad {\text { sense with your plot from part (a)? Give reasons for your }} \\ \quad {\text { answer. }}\end{array} \end{equation} $$f(x)=x^{2} \cos x, \quad x_{0}=\pi / 4$$

In Exercises \(35-40,\) write a differential formula that estimates the given change in volume or surface area. $$ \begin{array}{l}{\text { The change in the surface area } S=6 x^{2} \text { of a cube when the edge }} \\ {\text { lengths change from } x_{0} \text { to } x_{0}+d x}\end{array} $$
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