Find y'' by implicit differentiation.
5x2 + y2 = 3
y'' =?
Well, well, well, we have an implicit differentiation problem! To find y'', we'll have to do a little bit of math clowning around. So, are you ready for some mathematical laughter?
First, let's differentiate both sides of your equation with respect to x. Using the power rule for differentiation, we get:
d/dx(5x^2 + y^2) = d/dx(3)
Now, let's break it down one term at a time, shall we? For the left-hand side, the derivative of 5x^2 with respect to x is simply 10x. But what about the y^2 term? That's where the chain rule comes in!
The chain rule states that if we have a function of a function, we need to multiply the derivative of the outer function by the derivative of the inner function. In this case, the outer function is y^2 and the inner function is y itself.
So, when we differentiate y^2 with respect to x, we get 2y * (dy/dx). That's because we multiply the derivative of the outer function (2y) with the derivative of the inner function (dy/dx).
With that in mind, let's rewrite our equation:
10x + 2y * (dy/dx) = 0
Now, to find y'', we need to differentiate the equation once more with respect to x. This time, we'll be using the product rule for differentiation!
Differentiating 10x gives us 10, and differentiating 2y * (dy/dx) is where the product rule comes into play. The product rule states that the derivative of the product of two functions is the first function times the derivative of the second function, plus the second function times the derivative of the first function.
So, using the product rule, we get:
10 + 2(dy/dx) * (dy/dx) + 2y * (d^2y/dx^2) = 0
Now, we just need to solve this equation for y''. So, let's rearrange things a bit:
2(dy/dx) * (dy/dx) + 2y * (d^2y/dx^2) = -10
And now, let's solve for y'' by dividing both sides of the equation by 2y:
(d^2y/dx^2) = (-10 - 2(dy/dx) * (dy/dx)) / (2y)
And there you have it! Y'' is equal to (-10 - 2(dy/dx) * (dy/dx)) / (2y).
Now, I hope that explanation didn't make your head spin like a circus carousel, but if you have any more questions or need further clowning around, feel free to ask!
To find y'', we need to differentiate both sides of the equation with respect to x twice using the chain rule.
Given equation: 5x^2 + y^2 = 3
Differentiating equation with respect to x once:
d/dx (5x^2 + y^2) = d/dx (3)
10x + 2yy' = 0
Now, differentiate the equation obtained above with respect to x again:
d/dx (10x + 2yy') = d/dx (0)
10 + 2(y')^2 + 2yy'' = 0
Simplifying the equation:
2yy'' = -10 - 2(y')^2
yy'' = -5 - (y')^2
Therefore, the second derivative of y, y'', is given by yy'' = -5 - (y')^2.
To find the second derivative, y'', of the equation 5x^2 + y^2 = 3 using implicit differentiation, we need to follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
We'll differentiate each term separately using the chain rule and product rule, if necessary.
Differentiating the left side:
d/dx (5x^2) = 10x
Differentiating the right side:
For y^2, we need to use the chain rule. Let u = y(x).
d/dx (u^2) = 2u * du/dx
Step 2: Determine the derivative of y with respect to x, dy/dx.
To do this, rearrange the given equation and solve for dy/dx:
5x^2 + y^2 = 3
Differentiating both sides with respect to x:
10x + 2u * du/dx = 0
10x = -2u * du/dx
dy/dx = du/dx = -10x / (2u)
Step 3: Differentiate the obtained derivative, dy/dx, with respect to x to find the second derivative, d^2y/dx^2 or y''.
Differentiating dy/dx = -10x / (2u) with respect to x:
d/dx (dy/dx) = d/dx (-10x / (2u))
d^2y/dx^2 = -10 / (2u) - (-10x * (2u)' / (2u)^2)
Simplifying further using the chain rule:
d^2y/dx^2 = -10 / (2u) + (20x * du/dx) / (2u)^2
However, we also have the relation: y^2 = 3 - 5x^2 (from the original equation).
So substituting u = y(x) and rearranging the equation:
u^2 = 3 - 5x^2
2u * du/dx = -10x
du/dx = -10x / (2u)
Substituting this in the equation for the second derivative:
d^2y/dx^2 = -10 / (2u) + (20x * (-10x / (2u))) / (2u)^2
Simplifying further:
d^2y/dx^2 = -10 / (2u) - (100x^2) / (4u^3)
d^2y/dx^2 = -5 / u - (25x^2) / (2u^3)
Since we expressed u in terms of y, we can rewrite this as:
d^2y/dx^2 = -5 / y - (25x^2) / (2y^3)
Therefore, the second derivative, y'', of the equation 5x^2 + y^2 = 3 is given by:
y'' = -5 / y - (25x^2) / (2y^3)
that would be 5x^2 + y^2 = 3
for y' or dy/dx
10x + 2y y' = 0
y' = -10x/2y = -5x/y
for y''
y'' = (y(-5) - 5x y')/y^2
= (-5y - 5x(-5x/y)/y^2
= -5(y - 5x^2/y)/y^2
= -5(y^2 - 5x^2)/y^3