If 2 x^2+ 3 x + xy = 5 and y( 5 ) = -12, find y'( 5 ) by implicit differentiation.
4x dx + 3dx+ y dx + x dy=0
4x dx + 3dx+ y dx =- x dy
dy/dx= -4-3/x-y
y'=-4-3/5+12
check that.
i'm sorry but i don't know how you got that number did u get ((-4)-3)/(5+12) or something else?
To find y'(5) by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
Differentiating the left side:
d/dx(2x^2 + 3x + xy) = d/dx(5)
Using the power rule, the derivative of x^n is nx^(n-1), where n is a constant:
d/dx(2x^2) + d/dx(3x) + d/dx(xy) = 0
Differentiating each term:
4x + 3 + (x*d/dx(y) + y*d/dx(x)) = 0
Simplifying:
4x + 3 + (xy' + y) = 0
Step 2: Solve for y'.
Rearranging the equation:
xy' + y = -4x - 3
Step 3: Substitute the given value y(5) = -12.
Plugging in x = 5 and y = -12 into the equation:
(5)(y') + (-12) = -4(5) - 3
Simplifying:
5y' - 12 = -20 - 3
5y' - 12 = -23
Step 4: Solve for y'.
Adding 12 to both sides:
5y' = -23 + 12
5y' = -11
Dividing both sides by 5:
y' = -11/5
Therefore, y'(5) = -11/5.
To find y'(5) by implicit differentiation, we need to differentiate both sides of the equation with respect to x.
Let's differentiate each term step by step:
For the term 2x^2, the derivative is 4x, using the power rule.
For the term 3x, the derivative is 3, using the power rule.
For the term xy, we need to use the product rule. The derivative with respect to x is y + x(dy/dx).
For the constant term on the right side, the derivative is 0 since it doesn't involve x.
Now, let's put it all together:
4x + 3 + y + x(dy/dx) = 0
Rearranging the equation to solve for dy/dx, we get:
x(dy/dx) = -(4x + 3 + y)
Now, let's substitute x = 5 and y(5) = -12 into the equation:
5(dy/dx) = -(4(5) + 3 + (-12))
5(dy/dx) = -(20 + 3 - 12)
5(dy/dx) = -11
Finally, to find dy/dx, we divide both sides by 5:
dy/dx = -11/5
Therefore, y'(5) = -11/5.