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Chapter 3: Problem 21

Find \(d y / d x\) for the following functions. $$y=x \sin x$$

Short Answer

Expert verified
Answer: The derivative of the function \(y = x\sin x\) is \(\frac{dy}{dx} = \sin x + x\cos x\).

Step by step solution

01

Differentiate first function (u) with respect to x

To find the derivative of u with respect to x, we need to differentiate \(u(x)=x\) with respect to x. The derivative of x with respect to x is 1. Therefore, \(u'(x) = 1\).
02

Differentiate second function (v) with respect to x

Now, we need to find the derivative of the second function, \(v(x) = \sin x\), with respect to x. The derivative of \(\sin x\) with respect to x is \(\cos x\). Therefore, \(v'(x) = \cos x\).
03

Apply the product rule

Now, we will apply the product rule for differentiation which states that \((uv)' = u'v + uv'\). We have already found \(u'(x) = 1\) and \(v'(x) = \cos x\). Now, we can plug these values into the product rule formula to find the derivative of the given function. $$(x\sin x)' = (1)(\sin x) + (x)(\cos x)$$
04

Simplify the expression

By simplifying the expression, we get: $$\frac{dy}{dx} = \sin x + x\cos x$$ The derivative of the function \(y = x\sin x\) is \(\frac{dy}{dx} = \sin x + x\cos x\).

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Most popular questions from this chapter

Consider the following functions (on the given interval, if specified). Find the inverse function, express it as a function of \(x,\) and find the derivative of the inverse function. $$f(x)=\frac{x}{x+5}$$

Derivatives and inverse functions Find the slope of the curve \(y=f^{-1}(x)\) at (4,7) if the slope of the curve \(y=f(x)\) at (7,4) is \(\frac{2}{3}\)

Vertical tangent lines a. Determine the points at which the curve \(x+y^{3}-y=1\) has a vertical tangent line (see Exercise 52 ). b. Does the curve have any horizontal tangent lines? Explain.

Visualizing tangent and normal lines a. Determine an equation of the tangent line and normal line at the given point \(\left(x_{0}, y_{0}\right)\) on the following curves. (See instructions for Exercises \(63-68 .)\) b. Graph the tangent and normal lines on the given graph. $$\begin{aligned}&3 x^{3}+7 y^{3}=10 y\\\&\left(x_{0}, y_{0}\right)=(1,1)\end{aligned}$$

Given the function \(f,\) find the slope of the line tangent to the graph of \(f^{-1}\) at the specified point on the graph of $$f(x)=x^{3} ;(8,2)$$
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