if 2x^2+2x+ xy =2 and y(2)=-5 find y'(2) by implicit differentiation
Implicit differentiation:
(d/dx)(2x^2+2x+xy=2)
4x+2+y+xy'=0
y'(x)=-(4x+2+y(x))/x
y'(2)=-(4(2)+2+y(2))/2
=-(10-5)/2
=-5/2
To find y'(2) by implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x.
The derivative of 2x^2 + 2x + xy with respect to x is:
d/dx(2x^2 + 2x + xy) = d/dx(2) = 0 + 0 + y(dy/dx) = 0
Step 2: Simplify the equation obtained from differentiation.
Simplifying the equation will give us:
y(dy/dx) = 0
Step 3: Solve for dy/dx.
Divide both sides of the equation by y:
dy/dx = 0 / y
Step 4: Substitute the given value of y(2) = -5.
Substituting y = -5 into the equation, we have:
dy/dx = 0 / (-5)
Step 5: Simplify the expression.
dy/dx = 0
Therefore, y'(2) = 0.
To find y'(2) using implicit differentiation, we need to differentiate both sides of the equation with respect to x.
Given equation: 2x^2 + 2x + xy = 2
Differentiating both sides with respect to x:
d/dx (2x^2 + 2x + xy) = d/dx (2)
To differentiate the left-hand side, we use the product rule. For the right-hand side, the derivative of a constant is zero.
Differentiating 2x^2 using the power rule:
d/dx (2x^2) = 4x
Differentiating 2x using the power rule:
d/dx (2x) = 2
For the term xy, we need to apply the product rule. The derivative of x with respect to x is 1, and the derivative of y with respect to x is y' (since y is a function of x).
Using the product rule:
d/dx (xy) = x * (dy/dx) + y * (d/dx(x))
Simplifying the expression:
d/dx (xy) = x * (dy/dx) + y
Putting it all together:
4x + 2 + x(dy/dx) + y = 0
Now we can substitute the value of x = 2, and the given value y(2) = -5.
4(2) + 2 + 2(dy/dx) - 5 = 0
Simplifying the equation:
8 + 2 + 2(dy/dx) - 5 = 0
10 + 2(dy/dx) - 5 = 0
2(dy/dx) = -10 + 5
2(dy/dx) = -5
dy/dx = -5/2
Therefore, y'(2) = -5/2.