eNotes Homework Help is a way for educators to help students understand their school work. Our experts are here to answer your toughest academic questions! Once it’s posted to our site, your question could help thousands of other students.
Popular Titles

LiteratureThe outcome of the Tom Robinson trial drastically affects various members of Maycomb's community. Jem, Scout, and Dill lose their childhood innocence after witnessing racial injustice firsthand....

BusinessThe unemployment rate for this community would be 14.3%. Let us see how we arrive at this figure. As you can see in the link below, an economy’s unemployment rate is generally defined as the...

Math`F(x) =int_0^(e^(2x)) ln(t+1)dt` `F'(x)=?` Take note that if the function has a form `F(x) = int_a^(u(x)) f(t)dt` its derivative is `F'(x)=f(u(x))*u'(x)` Applying this formula, the derivative of...

Math`F(x)=int_pi^(lnx) cos(e^t)dt` `F'(x)=?` Take note that if the function has a form `F(x)=int_a^(u(x)) f(t)dt` its derivative is `F'(x)=f(u(x))*u'(x)` Applying this formula, the derivative of the...

MathFind the derivative if `y=e^(2x)tan(2x) ` : Let u=2x and rewrite as: `y=e^utanu ` Use the product rule noting that `d/(du)e^u=e^udu,d/(du)tanu=sec^2udu ` : `(dy)/(du)=e^udutanu+e^usec^2udu ` Since...

MathFind the derivative if `y=e^x(sinx+cosx) ` : Use the product rule: `(dy)/(dx)=e^(x)(cosxsinx)+e^(x)(sinx+cosx) ` `(dy)/(dx)=2e^xcosx `

MathFind the derivative if `y=(e^(2x))/(e^(2x)+1) ` : Use the quotient rule to get: `(dy)/(dx)=((e^(2x)+1)(2e^(2x))(e^(2x)*2e^(2x)))/(e^(2x)+1)^2 ` `(dy)/(dx)=(2e^(4x)+2e^(2x)2e^(4x))/(e^(2x)+1)^2 `...

MathFind the derivative if ` y=(e^x+1)/(e^x1) ` : Use the quotient rule to get: `(dy)/(dx)=((e^x1)(e^x)(e^x+1)(e^x))/(e^x1)^2 ` `(dy)/(dx)=(e^(2x)e^xe^(2x)e^x)/(e^x1)^2 ` `...

MathWe are asked to differentiate `y=(e^xe^(x))/2 ` : We use the fact that if f(x), g(x) are differentiable functions of x then `d/(dx)(f(x)+ g(x))=d/(dx)f(x)+d/(dx)g(x) ` and ` d/(dx)e^u=e^u...

MathFind the derivative for `y=2/(e^x+e^(x)) ` : Using the quotient rule we get: `(dy)/(dx)=(2(e^xe^(x)))/(e^x+e^(x))^2 `

MathFind the derivative of `y=ln((1+e^x)/(1e^x)) ` : Use a property of the natural logarithm to rewrite as: `y=ln(1+e^x)ln(1e^x) ` If u is a differentiable function of x, then ` d/(dx)ln(u)=(du)/u `...

MathFind the derivative if ` y=ln(1+e^(2x)) ` : If u is a differentiable function of x then ` d/(dx)lnu=(u')/u, d/(dx)e^(u)=e^u *u' ` so we get: `(dy)/(dx)=(2e^(2x))/(1+e^(2x)) `

MathFind the derivative if `y=e^(3/t^2) ` : If u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx) ` . Note that `d/(dt) 3/t^2=6/t^3 ` so we get: `(dy)/(dx)=6/t^3e^(3/t^2) `

MathFind the derivative if `y=(e^(t)+e^t))^3 ` : Use the power rule ( `d/(dx)u^n=n*u^(n1)(du)/(dx) ` where u is a differentiable function of x) to get: `(dy)/(dt)=3(e^(t)+e^t)^2(e^(t)+e^t) ` If...

MathFind the derivative if ` y=x^2e^(x)` : Use the product rule to get: `(dy)/(dx)=x^2(e^(x))+2xe^(x) ` `(dy)/(dx)=e^(x)(2xx^2) `

MathFind the derivative if `y=x^3e^x ` : Use the product rule to get: `(dy)/(dx)=x^3e^x+3x^2e^x ` `(dy)/(dx)=e^x(x^3+3x^2) `

MathFind the derivative if `y=xe^(4x) ` : If u is a differentiable function of x then ` d/(dx)e^u=e^u (du)/(dx) ` , so using the product rule we get: `(dy)/(dx)=x(4)e^(4x)+e^(4x) `...

MathFind the derivative if ` y=e^xlnx ` : If u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx), d/(dx)lnu=((du)/(dx))/u ` , so using the product rule we get: `(dy)/(dx)=e^xlnx+e^x*1/x...

MathFind the derivative if `y=5e^(x^2+5) ` : Note that if u is a differentiable function of x then `d/(dx)e^u=e^u (du)/(dx) ` , so using the constant rule we get: ` (dy)/(dx)=5(2x)e^(x^2+5) `...

MathFind the derivative if ` y=e^(x4) ` : If u is a differentiable function of x, then `d/(dx)e^u=e^u (du)/(dx) ` . Here u=x4 and du=1 so: `(dy)/(dx)=e^(x4) `

Math`y=e^(2x^3)` Find the derivative using the chain rule. `y'=e^(2x^3)*(6x^2)` `y'=6x^2e^(2x^3)` The answer is `6x^2e^(2x^3).`

Math`y=e^sqrtx` `y'=e^sqrt(x)*1/2x^(1/2)` `y'=e^sqrt(x)/[2sqrt(x)]` `y'=[sqrt(x)e^sqrt(x)]/[2x]` The derivative is `[sqrt(x)e^sqrt(x)]/[2x]`

Math`y=e^(8x)` `y'=e^(8x)*(8)` `y'=8e^(8x)` The answer is `8e^(8x)`

Math`f(x)=e^(2x)` Find the derivative using the Chain Rule. `f'(x)=e^(2x)*2` `f'(x)=2e^(2x)` The answer is `2e^(2x)`

MathWe are asked to show that the function `y=ln(4x) ` has an inverse by showing, using the derivative, that the function is monotonic on its entire domain. The domain for the function is x>0....

MathWe are asked to determine if ` ln(x^2) ` has an inverse; there is an inverse if the function is monotonic on its entire domain, so we use the derivative to determine the monotonicity:...

MathWe are asked to use the derivative to determine if ` y=lnx ` is monotonic on its domain. The domain of the natural logarithm is x>0: `y'=1/x ` and `1/x > 0 forall x>0 ` so the function is...

MathWe are asked to use the derivative of ` y=5000/(1+e^(2x)) ` to determine if the function has an inverse. (We can show that an inverse exists if the function is monotonic on its entire domain.)...

Math`e^(lnx)=4` To solve, apply the logarithm rule `e^(lnm) = m` . So the left side simplifies to: `x=4` Therefore, the solution of the given equation is x=4.

MathWe are asked to show that `f(x)=x+2,[2,oo) ` has an inverse by showing that the function is monotonic on the interval using the derivative: By definition, ` f(x)=x+2={[x+2,x+2...

MathGiven `f(x)=(x4)^2, [4,oo) ` we are asked to show that the function has an inverse since it is monotonic on the given interval: `f'(x)=2(x4) ` and `2(x4)>=0 forall x>=4 ` so the function...

Math`f(x)=2xx^2` `f'(x)=12x=0` `12x=0` `2x=1` `x=1/2` A critical value for x is `x=1/2.` In the interval `(oo, 1/2)` the function increases. In the interval `(1/2,oo)` the function...

HistoryThe Civil Rights Movement of the 1960s were largely aimed at exposing the injustices AfricanAmericans faced in society, while the social justice movements of the 1970s concentrated on agitating...

HistoryOne event from the 1850s that heightened tensions between the North and the South was the raid by the abolitionist John Brown on the federal armory at Harpers Ferry, which was then in the state of...

ReferenceAlthough slavery has ended and white supremacy is no longer encouraged in the United States, the truth is that the history of white supremacy still has a strong lingering impact on the US and...

Native American LiteratureThe writer is apprehensive and a little afraid of how he will be received when he first arrives in Woodruff. This is because the writer has presumably been away for a while. He hasn't seen his...

Patrick HenryIn the past, "awful" meant "important." What does Henry mean by the word "awful" in his speech to...Patrick Henry wanted to raise a militia to prevent the British from taking advantage of them. In this impassioned speech, Henry’s point was that they could not stand back and let the British walk...

A Christmas CarolHow do the Cratchits react to their Christmas feast, and what does their celebration show Scrooge...The Cratchit family is grateful for their feast even though it is meager, and Scrooge realizes that you do not need much to be happy as long as you have people you love. The Cratchit family reminds...

ArabyThe most obvious character who is not named in James Joyce's "Araby" is Mangan's sister. Throughout the story, she is known as "Mangan's sister," not as a named character despite the fact that the...

FrindleWhen the novel begins, Nick is a discontented teen looking for ways to cause minor trouble in his Language Arts class. He finds a way by calling a pen “frindle,” never imagining how far this...

A White HeronMotifs are recurring symbols which support the themes or main ideas in any story. In A White Heron, the young sportsman's gun is a recurring motif. To Sylvia, the gun represents strange emotions...

Public HealthBased on your background in public health, here are some headings I would recommend for use in promotional materials regarding the Rotavirus: 1) What the Rotavirus is and how it affects the health...

The Devil's HighwayIn any game, there are always two opposing sides and none more so than in the area of illegal immigration. In the book, Urrea discusses how the Border Patrol agents (La Migra) often find themselves...

The PrinceIn The Prince and Discourses on Livy, Machiavelli exhibits the characteristics of humanism in many ways. First, both works are full of allusions to classical politics, especially the Roman...

To AutumnThe theme of "To Autumn" is the transitory quality of nature. In one of the marvelous letters written by Keats to friends, he explained that he composed "To Autumn" because Somehow a stubble plain...

ScienceThe laws of thermodynamics are as follows. The first law is that the increase in energy in a system is the sum of energy added to the system. This means that a system will not change energy unless...

Ode to a NightingaleIn the poem, the speaker claims to be "half in love" with death. He thinks that death will release him from his mental torment. Indeed, he's portrayed death in a soft light in "many a mused rhyme"...

War Is KindHow does the poet describe both negative and positive aspects of war without being partial to any...The poet Stephen Crane is being intentionally ironic. He does not really believe that war is kind but that it is a horrible and senseless thing. The Summary in the enotes study guide for the poem...

ReferenceChildren of preschool age (23) often begin talking about what they dreamt about, though these reports usually involve static pictures rather than episodes with characters or emotions. Children...

Oedipus