Find the 130th term of the sequence

If x^2y-3x=y^3-3, then at the point (-1,2), (dy/dx)?

zavuch27 [327]

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 dy
Find —— for an implicit function:
 dx

x²y – 3x = y³ – 3

First, differentiate implicitly both sides with respect to x. Keep in mind that y is not just a variable, but it is also a function of x, so you have to use the chain rule there:

$\mathsf{\dfrac{d}{dx}(x^2 y-3x)=\dfrac{d}{dx}(y^3-3)}\\\\\\
\mathsf{\dfrac{d}{dx}(x^2 y)-3\,\dfrac{d}{dx}(x)=\dfrac{d}{dx}(y^3)-\dfrac{d}{dx}(3)}$

Applying the product rule for the first term at the left-hand side:

$\mathsf{\left[\dfrac{d}{dx}(x^2)\cdot y+x^2\cdot \dfrac{d}{dx}(y)\right]-3\cdot 1=3y^2\cdot \dfrac{dy}{dx}-0}\\\\\\
\mathsf{\left[2x\cdot y+x^2\cdot \dfrac{dy}{dx}\right]-3=3y^2\cdot \dfrac{dy}{dx}}$

 dy
Now, isolate —— in the equation above:
 dx

$\mathsf{2xy+x^2\cdot \dfrac{dy}{dx}-3=3y^2\cdot \dfrac{dy}{dx}}\\\\\\
\mathsf{2xy+x^2\cdot \dfrac{dy}{dx}-3-3y^2\cdot \dfrac{dy}{dx}=0}\\\\\\
\mathsf{x^2\cdot \dfrac{dy}{dx}-3y^2\cdot \dfrac{dy}{dx}=-\,2xy+3}\\\\\\
\mathsf{(x^2-3y^2)\cdot \dfrac{dy}{dx}=-\,2xy+3}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{-\,2xy+3}{x^2-3y^2}\qquad\quad for~~x^2-3y^2\ne 0}$

Compute the derivative value at the point (– 1, 2):

x = – 1 and y = 2

$\mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=\dfrac{-\,2\cdot (-1)\cdot 2+3}{(-1)^2-3\cdot 2^2}}\\\\\\
\mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=\dfrac{4+3}{1-12}}\\\\\\
\mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=\dfrac{7}{-11}}\\\\\\\\ \therefore~~\mathsf{\left.\dfrac{dy}{dx}\right|_{(-1,\,2)}=-\,\dfrac{7}{11}}\quad\longleftarrow\quad\textsf{this is the answer.}$

I hope this helps. =)

Tags: implicit function derivative implicit differentiation chain product rule differential integral calculus

6 0

2 years ago

Divide. −5 1/3÷2 3/4 −12 5/6 −7 5/12 −1 31/33 −1 11/12

Dvinal [7]

Answer:

-26.0454545455

Step-by-step explanation:

4 0

2 years ago

What is the value of ? 2 3 4 5

Fed [463]

Answer:

idk what your asking but assuming u mean multiplying the answer is 120

Step-by-step explanation:

8 0

2 years ago

MONA LISA A visitor to the Louvre Museum in Paris wants to sketch a drawing of the Mona Lisa, a famous painting. The original pa

nydimaria [60]

You need to find a scale factor of dilation (probably contraction) to make the replica.

Given that the units of the original painting are centimeters and the units of the sheet of paper are in inches, you have to convert one of the units.

I choose to covert inches to centimeters.

To do that, use the conversion factor 1 = 2.54 cm / inch

=> 8.5 in * [2.54 cm/in] = 21.59 cm

=> 11 in * [2.54 cm/in] = 27.94 cm

Now determine the scale factor to convert 77 cm to 27.94 cm and 53 cm to 21.59 cm. That just require use of division operation:

77 / 27.94 = 2.76

53 / 21.59 = 2.45

Then the higher scale factor is the relevant one, and you have to reduce the original painting by a factor of (1 / 2.76) which will lead to fit in the sheet of paper.

7 0

3 years ago

What is 0.2401 as a fraction

vlabodo [156]

Answer:

24/100 or 6/25

Step-by-step explanation:

.24 is the same as 24/100 if you need to simplify then just find a common number to divide by.

Hope this helps! Please mark Brainliest! Have a great day!

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3 0

3 years ago

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