If 4x^(2)+5x+xy=4 and y(4)=-20, find y'(4) by implicit differentiation.
I got y'=(-8x-5-y)/(x)
Differentiate the first equation with respect to x, with y being treated as a function of x.
8x + 5 + y + x dy/dx = 0
dy/dx = -8 -5/x -y/x
(That agrees with your formula).
The want the actual value at x=4.
When x=4, 64 + 20 + 4y = 4, so
80 + 4y = 0, and
y (@x=4)= -20
dy/dx (@x=4) = -8 -5/4 +20/4 = -17/4
Oh, I see you're dabbling in implicit differentiation, huh? Well, it looks like you're on the right track with your differentiation. However, I'm feeling a little humor-bug in my circuits, so how about we try a more amusing approach?
Let's see...imagine you're at a circus, and you're riding a unicycle while juggling equations. Suddenly, a clown rides up on a mini tricycle and says, "Hey, I can help you find y'(4)!" You nod enthusiastically, ready for some comic relief.
The clown jumps off the tricycle, takes your equation, and starts juggling numbers and variables in the air. While juggling, he mutters, "Ah, let's find the derivative, shall we? We take the derivative of both sides with respect to x."
After a few drops and mishaps, he proudly presents the derivative equation, saying, "Tada! We've got y'(4) = (-8x - 5 - y) / x. And remember kids, always watch out for those clowns who juggle equations, they might bring a little chaos to your math party!"
So there you have it, according to the skilled clown, y'(4) = (-8x - 5 - y) / x. Just be careful not to bump into any more equation-juggling clowns on your math journey!
To find y'(4) by implicit differentiation, we need to differentiate both sides of the equation with respect to x.
Let's differentiate the equation 4x^2 + 5x + xy = 4 term by term.
Differentiating the left side:
The derivative of 4x^2 with respect to x is 8x.
The derivative of 5x with respect to x is 5.
To differentiate xy with respect to x, we need to use the product rule:
The derivative of x with respect to x is 1, so we get x * d(y)/dx + y.
Differentiating the right side:
The derivative of 4 with respect to x is 0 since it is a constant.
Putting it all together, we have:
8x + 5 + x * (dy/dx) + y = 0
Now, let's substitute y = -20 and x = 4 into the equation and solve for dy/dx.
8(4) + 5 + 4 * (dy/dx) + (-20) = 0
32 + 5 + 4 * (dy/dx) - 20 = 0
37 + 4 * (dy/dx) = 20
4 * (dy/dx) = -17
dy/dx = -17/4
Therefore, y'(4) = -17/4.
To find y'(4) by implicit differentiation, we first need to differentiate both sides of the equation 4x^2 + 5x + xy = 4 with respect to x.
Let's differentiate each term on the left-hand side using the rules of differentiation:
Differentiating 4x^2 with respect to x gives us:
d/dx (4x^2) = 8x
Differentiating 5x with respect to x gives us:
d/dx (5x) = 5
To differentiate the term xy, we need to use the product rule, which states that d/dx (uv) = u'v + uv', where u and v are functions of x.
In this case, u = x and v = y. Differentiating xy with respect to x gives us:
d/dx (xy) = x * dy/dx + y * dx/dx = x * dy/dx + y
Now, let's put all the derivatives together and rearrange the terms:
8x + 5 + x * dy/dx + y = 0
To solve for dy/dx, we isolate it on one side of the equation:
x * dy/dx = -8x - 5 - y
Finally, divide both sides of the equation by x:
dy/dx = (-8x - 5 - y) / x
To find y'(4), substitute x = 4 into the equation:
y'(4) = (-8(4) - 5 - y) / 4
Since we are given that y(4) = -20, substitute this value into the equation:
y'(4) = (-8(4) - 5 - (-20)) / 4
Simplifying this expression, we get:
y'(4) = (-32 - 5 + 20) / 4
= -17/4
Therefore, y'(4) = -17/4.