1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Amiraneli [1.4K]
2 years ago
6

(d). Use an appropriate technique to find the derivative of the following functions:

Mathematics
1 answer:
natima [27]2 years ago
3 0

(i) I would first suggest writing this function as a product of the functions,

\displaystyle y = fgh = (4+3x^2)^{1/2} (x^2+1)^{-1/3} \pi^x

then apply the product rule. Hopefully it's clear which function each of f, g, and h refer to.

We then have, using the power and chain rules,

\displaystyle \frac{df}{dx} = \frac12 (4+3x^2)^{-1/2} \cdot 6x = \frac{3x}{(4+3x^2)^{1/2}}

\displaystyle \frac{dg}{dx} = -\frac13 (x^2+1)^{-4/3} \cdot 2x = -\frac{2x}{3(x^2+1)^{4/3}}

For the third function, we first rewrite in terms of the logarithmic and the exponential functions,

h = \pi^x = e^{\ln(\pi^x)} = e^{\ln(\pi)x}

Then by the chain rule,

\displaystyle \frac{dh}{dx} = e^{\ln(\pi)x} \cdot \ln(\pi) = \ln(\pi) \pi^x

By the product rule, we have

\displaystyle \frac{dy}{dx} = \frac{df}{dx}gh + f\frac{dg}{dx}h + fg\frac{dh}{dx}

\displaystyle \frac{dy}{dx} = \frac{3x}{(4+3x^2)^{1/2}} (x^2+1)^{-1/3} \pi^x - (4+3x^2)^{1/2} \frac{2x}{3(x^2+1)^{4/3}} \pi^x + (4+3x^2)^{1/2} (x^2+1)^{-1/3} \ln(\pi) \pi^x

\displaystyle \frac{dy}{dx} = \frac{3x}{(4+3x^2)^{1/2}} \frac{1}{(x^2+1)^{1/3}} \pi^x - (4+3x^2)^{1/2} \frac{2x}{3(x^2+1)^{4/3}} \pi^x + (4+3x^2)^{1/2} \frac{1}{ (x^2+1)^{1/3}} \ln(\pi) \pi^x

\displaystyle \frac{dy}{dx} = \boxed{\frac{\pi^x}{(4+3x^2)^{1/2} (x^2+1)^{1/3}} \left( 3x - \frac{2x(4+3x^2)}{3(x^2+1)} + (4+3x^2)\ln(\pi)\right)}

You could simplify this further if you like.

In Mathematica, you can confirm this by running

D[(4+3x^2)^(1/2) (x^2+1)^(-1/3) Pi^x, x]

The immediate result likely won't match up with what we found earlier, so you could try getting a result that more closely resembles it by following up with Simplify or FullSimplify, as in

FullSimplify[%]

(% refers to the last output)

If it still doesn't match, you can try running

Reduce[<our result> == %, {}]

and if our answer is indeed correct, this will return True. (I don't have access to M at the moment, so I can't check for myself.)

(ii) Given

\displaystyle \frac{xy^3}{1+\sec(y)} = e^{xy}

differentiating both sides with respect to x by the quotient and chain rules, taking y = y(x), gives

\displaystyle \frac{(1+\sec(y))\left(y^3+3xy^2 \frac{dy}{dx}\right) - xy^3\sec(y)\tan(y) \frac{dy}{dx}}{(1+\sec(y))^2} = e^{xy} \left(y + x\frac{dy}{dx}\right)

\displaystyle \frac{y^3(1+\sec(y)) + 3xy^2(1+\sec(y)) \frac{dy}{dx} - xy^3\sec(y)\tan(y) \frac{dy}{dx}}{(1+\sec(y))^2} = ye^{xy} + xe^{xy}\frac{dy}{dx}

\displaystyle \frac{y^3}{1+\sec(y)} + \frac{3xy^2}{1+\sec(y)} \frac{dy}{dx} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} \frac{dy}{dx} = ye^{xy} + xe^{xy}\frac{dy}{dx}

\displaystyle \left(\frac{3xy^2}{1+\sec(y)} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} - xe^{xy}\right) \frac{dy}{dx}= ye^{xy} - \frac{y^3}{1+\sec(y)}

\displaystyle \frac{dy}{dx}= \frac{ye^{xy} - \frac{y^3}{1+\sec(y)}}{\frac{3xy^2}{1+\sec(y)} - \frac{xy^3\sec(y)\tan(y)}{(1+\sec(y))^2} - xe^{xy}}

which could be simplified further if you wish.

In M, off the top of my head I would suggest verifying this solution by

Solve[D[x*y[x]^3/(1 + Sec[y[x]]) == E^(x*y[x]), x], y'[x]]

but I'm not entirely sure that will work. If you're using version 12 or older (you can check by running $Version), you can use a ResourceFunction,

ResourceFunction["ImplicitD"][<our equation>, x]

but I'm not completely confident that I have the right syntax, so you might want to consult the documentation.

You might be interested in
Help plz, these questions are very confusing :/
padilas [110]

Answer:

1.

cos ∅=b/h=24/25

:.b=24

h=25

by using Pythagoras law

h²=p²+b²

25²=p²+24²

p²=25²-24²

p=√7²=7

now

sin ∅=P/h=7/25

tan∅=p/b=7/24

2.cos ∅ is your answer

sec∅-sin∅tan∅

1/cos∅-sin²/cos∅

(1-sin²∅)/cos∅

cos²∅/cos∅

cos∅

3.

tan²∅-sec²x

sin²∅/cos²∅-¹/cos²∅

(sin²∅-1)/cos²∅

8 0
2 years ago
The fee for charging an electric car at station A is $2 to start and $4 for each hour or fraction of an hour. Which point is NOT
Alex777 [14]
I think it is D


I think ok
8 0
3 years ago
15 · 20 = 12y ??<br> 180° = 74° + C ??
Vsevolod [243]

Answer:

answer is 90 degree I think so

4 0
3 years ago
The image represents a rectangular patio that has a length equal to 4 feet more than the width. What is the perimeter of the pat
IrinaK [193]

Answer:

  4w+8 feet

Step-by-step explanation:

The perimeter of a rectangle is the sum of the lengths of its sides. Since opposite sides are identical in length, it can be computed as double the sum of adjacent side lengths. Here, the lengths are in feet.

  P = 2(L+W)

  P = 2((w+4) +w)

  P = 4w +8

The perimeter of the patio is 4w+8 feet.

4 0
2 years ago
Need help with this Question
vazorg [7]

Answer:

Step-by-step explanation:

W∩ X is asking for the intersection of sets W and X, or simply, which numbers can be found in both sets.

The numbers shared between the two sets are 3, 4, 5.

Next, we need to union this result with the set Y, or simply, combine the numbers in both sets together.

This would result in 2, 3, 4, 5

5 0
3 years ago
Other questions:
  • Aimee must add together the following numbers:3.50 + 4.00 + (−1.25) + (−7.50) + 5.25 + 2.00 As her first step, Aimee writes:(3.5
    10·2 answers
  • From 1,126 meters above sea level, Dedra took off in her helicopter and ascended 228 meters. What is Dedra's elevation now?
    15·1 answer
  • Find the difference in the volume and total area of a cylinder with both a radius and height of1 r-1, h-1
    10·1 answer
  • Which sentences are punctuated correctly? Select 2 options.
    5·2 answers
  • For a binomial probability distribution, it is unusual for the number of successes to be less than μ − 2.5σ or greater than μ +
    5·1 answer
  • Find the value of the trigonometric ratio. Make sure to simplify the fraction if needed.
    10·1 answer
  • A ruler cost x pence. A pen costs 10 pence more than the ruler. Write an expression, in terms of x, for the cost of a pen .
    14·1 answer
  • Write an arithmetic sequence that has a common difference of -3 and whose
    8·1 answer
  • What is a number that is 450 rounded to the nearest 10th
    12·1 answer
  • The World Health Organization lists the following overall literacy rates per 100 people for countries. Which answer choice inclu
    10·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!