Please help, it would really make my day! I’ll give branliest if it’s correct! :-)

Use implicit differentiation to find the slope of the tangent line at the given point:

Salsk061 [2.6K]

Answer:

$\frac{dy}{dx}=0$

Step-by-step explanation:

So we have the equation:

$(x^2+y^2)^2=4x^2y$

And we want to find the slope of the tangent line at the point (1,1).

So, let's implicitly differentiate. Take the derivative of both sides:

$\frac{d}{dx}[(x^2+y^2)^2]=\frac{d}{dx}[4x^2y]$

Let's do each side individually.

Left:

We can use the chain rule:

$(u(v(x))'=u'(v(x))\cdot v'(x)$

Let's let v(x) be x²+y². So, u(x) is x². Thus, the u'(x) is 2x. Therefore:

$\frac{d}{dx}[(x^2+y^2)^2]=2(x^2+y^2)(\frac{d}{dx}[x^2+y^2])$

We can differentiate x like normal. However, for y, we must differentiate implicitly. pretend y is y(x). This gives us:

$\frac{d}{dx}[(x^2+y^2)^2]=2(x^2+y^2)(\frac{d}{dx}[x^2]+\frac{d}{dx}[y^2(x)])$

Differentiate:

$\frac{d}{dx}[(x^2+y^2)^2]=2(x^2+y^2)(2x+2y\frac{dy}{dx})$

Therefore, our left side is:

$2(x^2+y^2)(2x+2y\frac{dy}{dx})$

Right:

We have:

$\frac{d}{dx}[4x^2y]$

Let's move the 4 outside:

$=4\frac{d}{dx}[x^2y]$

Use the product rule:

$=4(\frac{d}{dx}[x^2]y+x^2\frac{d}{dx}[y])$

Differentiate:

$=4(2xy+x^2\frac{dy}{dx})$

Therefore, our entire equation is:

$2(x^2+y^2)(2x+2y\frac{dy}{dx})=4(2xy+x^2\frac{dy}{dx})$

So, to find the derivative at (1,1), substitute 1 for x and 1 for y.

$2((1)^2+(1)^2)(2(1)+2(1)\frac{dy}{dx})=4(2(1)(1)+(1)^2\frac{dy}{dx})$

Evaluate.

$2((1)+(1))(2+2\frac{dy}{dx})=4(2+\frac{dy}{dx})$

Simplify. Also, let's distribute the right:

$2(2)(2+2\frac{dy}{dx})=8+4\frac{dy}{dx}$

Multiply.

$4(2+2\frac{dy}{dx})=8+4\frac{dy}{dx}$

Distribute the left:

$8+8\frac{dy}{dx}=8+4\frac{dy}{dx}$

Subtract 8 from both sides:

$8\frac{dy}{dx}=4\frac{dy}{dx}$

Subtract 4(dy/dx) from both sides:

$4\frac{dy}{dx}=0$

Divide both sides by 4:

$\frac{dy}{dx}=0$

Therefore, the slope at the point (1,1) is 0.

And we're done!

We can verify this using the graph. The slope of the line tangent to the point (1,1) seems like it would be horizontal, giving us a slope of 0.

Edit: Typo

5 0

4 years ago

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Is 1.4 greater than, less than, or equal to -14/7

alekssr [168]

It is greater than -14/7 because first of all, -14/7 is negative and 1.4 positive which means the positive is greater than negative.
also -14/7 is -2
1.4 is greater than -2

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4 years ago

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Solve the simultaneous equation.......2x + 10y = 50 (equation 1) and 1.5x + 25y = 90 (equation 2)

cricket20 [7]

Step-by-step explanation:

2x + 10y = 50 ... 1

1.5x + 25y = 90 ...2

...1 × 2.5

5x + 25y = 125 ...3

...3 - ...2

3.5x = 35

x = 10

replaced x = 10 in ...1

2(10) + 10y = 50

20 + 10y = 50

10y = 50 - 20

10y = 30

y = 3

6 0

3 years ago

MATH PLEASE HELP! WILL MARK BRAINLIEST ANSWER!

Nikolay [14]

I believe it is 3,4 4,5 3,6

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La gatita de Catalina a los 2 meses de vida tenía un peso de 0,85 kg. A los 12 meses posee un peso de 2,25 kg ¿Cuál fue el porce

Evgen [1.6K]

Answer:

can u pls post this question in English language

sorry can't understand

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3 years ago