i still understand this problem

If x^2/25 +y^2/49=1 and y(3)=5.6000, find y'(3) by implicit differentiation.

i mean i still don't understand this problem

I got that!

Use implicit differentiation, which is actually derived from the chain rule, except that when it comes to dy/dx, since y is not know, we simply write dy/dx, or y'.

df(y)/dx = df(y)/dy * dy/dx

x^2/25 +y^2/49=1
Differentiate both sides:
2x/25 + 2y/49 * dy/dx = 0
(dy/dx = y'(x))
y'(x)=-(2x/25)*(49/2y)
=-(49x/25y)
y(3)=5.6 => x=3, y=5.6
y'(3)= -(49(3))/(25(5.6))
= -147/140
= -21/20

thanks

You're welcome! :)

To find the derivative of y with respect to x, we can differentiate the equation x^2/25 + y^2/49 = 1 implicitly using the chain rule.

1. Start by differentiating both sides of the equation with respect to x:
d/dx(x^2/25) + d/dx(y^2/49) = d/dx(1)

2. The derivative of x^2/25 with respect to x is (2x)/25. Since x does not involve y, we only need to differentiate the x part.

3. For the term y^2/49, we need to apply the chain rule since y is a function of x. Let's define u = y(x). Then, the derivative of y^2/49 with respect to x is:
d/dx((y^2/49)) = d/dx((u^2)/49) = (2u/49) * du/dx

4. Now we need to find du/dx, which is the derivative of y with respect to x. We denote this as y'(x).

5. Putting it all together, the equation becomes:
(2x)/25 + (2u/49) * du/dx = 0

6. To find y'(x), we need to solve for du/dx, which can be done by rearranging the equation:
(2u/49) * du/dx = -(2x)/25
du/dx = (-(2x)/25) / (2u/49)
du/dx = -49x / (25u)

7. Now substitute u = y(x) back into the equation:
du/dx = -49x / (25y)

8. Finally, evaluate y'(3) by substituting x = 3 into the derived equation:
y'(3) = -49(3) / (25y(3))
y'(3) = -147 / (25 * 5.6000)
y'(3) = -147 / 140
y'(3) = -1.05