Graph the point (5, 0) on the coordinate plane.

Find gradient xe^y + 4 ln y = x² at (1, 1)

cricket20 [7]

$xe^y+4\ln y=x^2$

Differentiate both sides with respect to x, assuming y = y(x).

$\dfrac{\mathrm d(xe^y+4\ln y)}{\mathrm dx}=\dfrac{\mathrm d(x^2)}{\mathrm dx}$

$\dfrac{\mathrm d(xe^y)}{\mathrm dx}+\dfrac{\mathrm d(4\ln y)}{\mathrm dx}=2x$

$\dfrac{\mathrm d(x)}{\mathrm dx}e^y+x\dfrac{\mathrm d(e^y)}{\mathrm dx}+\dfrac4y\dfrac{\mathrm dy}{\mathrm dx}=2x$

$e^y+xe^y\dfrac{\mathrm dy}{\mathrm dx}+\dfrac4y\dfrac{\mathrm dy}{\mathrm dx}=2x$

Solve for dy/dx :

$e^y+\left(xe^y+\dfrac4y\right)\dfrac{\mathrm dy}{\mathrm dx}=2x$

$\left(xe^y+\dfrac4y\right)\dfrac{\mathrm dy}{\mathrm dx}=2x-e^y$

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2x-e^y}{xe^y+\frac4y}$

If y ≠ 0, we can write

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2xy-ye^y}{xye^y+4}$

At the point (1, 1), the derivative is

$\dfrac{\mathrm dy}{\mathrm dx}\bigg|_{x=1,y=1}=\boxed{\dfrac{2-e}{e+4}}$

4 0

3 years ago

Solve the equation given in Exercise 15 with and without using the LCD of the fractions. Are your answers the same?

Colt1911 [192]

Without LCD :
1/2(4x + 6) = 1/3(9x - 24)
2x + 3 = 3x - 8
2x - 3x = -8 - 3
-x = - 11
x = 11

with LCD :
1/2(4x + 6) = 1/3(9x - 24) ...multiply by 6
3(4x + 6) = 2(9x - 24)
12x + 18 = 18x - 48
12x - 18x = -48 - 18
-6x = - 66
x = -66/-6
x = 11

Yes, the answers are the same :)

3 0

3 years ago

(C) Write an equation for the amount, a, that Maria makes as a function of the number of hours, h, that she

Svetradugi [14.3K]

Answer:

yes

Step-by-step explanation:

yes

6 0

3 years ago

How does a person use credit to make a

Black_prince [1.1K]

Answer:

makes a promise to pay at some point in the future

Step-by-step explanation:

Using credit means you borrow money to buy something, which is called a "loan", then you can buy the things you want, and you can pay it back later (with interest).

6 0

3 years ago

Henry knows that the circumference of a circle is 18 pie inches. What is the area of the circle?

Aneli [31]

Answer:

254.34

Step-by-step explanation:

This is because A=C2

4π which is 18x18 divided 4x 3.14 which equals 254.34

6 0

2 years ago

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