Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 16: Problem 1

Find \(\frac{\mathrm{dy}}{\mathrm{d} x}\) where \(y\) is given by (a) \(x \cos x\) (b) \(x e^{x^{k}}\) (c) \(\sin x \cos 2 x\) (d) \(x^{3} \mathrm{e}^{2 x}\) (e) \(x^{4} \sin 2 x\)

Short Answer

Expert verified
Question: Find the first derivative of each function with respect to x. (a) \(y = x \cos x\) (b) \(y = x e^{x^k}\) (c) \(y = \sin x \cos 2x\) (d) \(y = x^3 e^{2x}\) (e) \(y = x^4 \sin 2x\) Solution: (a) \(y' = \cos x - x \sin x\) (b) \(y' = e^{x^k} (1 + kx^k)\) (c) \(y' = \cos x \cos 2x - 2\sin x \sin 2x\) (d) \(y' = e^{2x} (3x^2 + 2x^3)\) (e) \(y' = 4x^3 \sin 2x + 2x^4 \cos 2x\)

Step by step solution

01

(a) Differentiating \(x \cos x\)

To differentiate \(x \cos x\), we will use the product rule, which states that for two functions \(u(x)\) and \(v(x)\), \({d(uv)}/{dx} = u'v + uv'\). Here, \(u(x) = x\) and \(v(x) = \cos x\). First, find the derivatives of \(u(x)\) and \(v(x)\): \(u'(x) = \frac{d}{dx}(x) = 1\) \(v'(x) = \frac{d}{dx}(\cos x) = -\sin x\) Now, apply the product rule: \(\frac{dy}{dx} = u'v + uv' = (1)(\cos x) + (x)(-\sin x) = \cos x - x \sin x\)
02

(b) Differentiating \(x e^{x^k}\)

Again, we will use the product rule for \(u(x) = x\) and \(v(x) = e^{x^k}\). This time, we need the chain rule for \(v'(x)\), which states that if you have a composition of two functions \(g(f(x))\), then its derivative is \(g'(f(x))f'(x)\). Here, let \(g(x) = e^x\) and \(f(x) = x^k\). First, find the derivatives: \(u'(x) = \frac{d}{dx}(x) = 1\) \(v'(x) = \frac{d}{dx}(e^{x^k}) = e^{x^k} \frac{d}{dx}(x^k) = e^{x^k} (kx^{k-1})\) Apply the product rule: \(\frac{dy}{dx} = u'v + uv' = (1)( e^{x^k})+ (x)(e^{x^k} (kx^{k-1}))= e^{x^k} (1 + kx^k)\)
03

(c) Differentiating \(\sin x \cos 2x\)

Use the product rule with \(u(x) = \sin x\) and \(v(x) = \cos 2x\). We differentiate: \(u'(x) = \frac{d}{dx}(\sin x) = \cos x\) \(v'(x) = \frac{d}{dx}(\cos 2x) = -2 \sin(2x)\) Apply the product rule: \(\frac{dy}{dx} = u'v + uv' = (\cos x)(\cos 2x) + (\sin x)(-2 \sin 2x) = \cos x \cos 2x - 2\sin x \sin 2x\)
04

(d) Differentiating \(x^3 e^{2x}\)

Use the product rule with \(u(x) = x^3\) and \(v(x) = e^{2x}\). We differentiate: \(u'(x) = \frac{d}{dx}(x^3) = 3x^2\) \(v'(x) = \frac{d}{dx}(e^{2x}) = 2e^{2x}\) Apply the product rule: \(\frac{dy}{dx} = u'v + uv' = (3x^2)( e^{2x})+ (x^3)(2e^{2x})= e^{2x} (3x^2 + 2x^3)\)
05

(e) Differentiating \(x^4 \sin 2x\)

Use the product rule with \(u(x) = x^4\) and \(v(x) = \sin 2x\). We differentiate: \(u'(x) = \frac{d}{dx}(x^4) = 4x^3\) \(v'(x) = \frac{d}{dx}(\sin 2x) = 2\cos 2x\) Apply the product rule: \(\frac{dy}{dx} = u'v + uv' = (4x^3)(\sin 2x) + (x^4)(2\cos 2x) = 4x^3 \sin 2x + 2x^4 \cos 2x\)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) given (a) \(x^{2}+x^{3}+y^{2}-y^{3}=1\) (b) \(2 x^{2}-y^{2}+3 x y-7 x-10 y=0\) (c) \(x y^{2}+\frac{y}{x}=\mathrm{e}^{x}\) (d) \(\ln (x y)-\sqrt{x}=\sqrt{y}\) (e) \(x \sin y+y^{2} \cos 2 x=y\)

Find the equation of the tangent to \(y=x \mathrm{e}^{x}\) where \(x=1\).

Find \(y^{\prime}\) where \(y\) is given by (a) \(\frac{x^{2}+1}{2 x+3}\) (b) \(\frac{\cos 2 t}{\sin 3 t}\) (c) \(\frac{r^{3}}{3 e^{2 r}}\) (d) \(\frac{v+\sin v}{1+\mathrm{e}^{v}}\) (e) \(\frac{\sqrt{x}}{x+1}\)

Differentiate the following functions of \(y\) with respect to \(x\) : (a) \((y+3)^{4}\) (b) \(\left(y^{2}+3\right)^{4}\)

Find \(y^{\prime}\) given \(\left(x^{2}-y^{3}\right)^{6}=\mathrm{e}^{x y}\).
See all solutions

Recommended explanations on Physics Textbooks

Force

Read Explanation

Torque and Rotational Motion

Read Explanation

Further Mechanics and Thermal Physics

Read Explanation

Physical Quantities And Units

Read Explanation

Energy Physics

Read Explanation

Classical Mechanics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free