1. Consider the following equation.
xy + 7x + 5y = –8
(a) Differentiate both sides with respect to x to complete the following.
( blank )y ' + y + blank =0
(b)Solve for y ' in terms of x and y.
(c) Calculate dy/dx at the point (4,-4).
Can anyone help me on this questions I'm struggling so badly....
for xy + 7x + 5y = –8
use the product rule for the xy term
x(dy/dx) + y + 7 5dy/dx = 0
dy/dx(x+5) = -y-7
dy/dx = -(y+7)/(x+5)
sub in the point (4,-4)
which one is part a and part b.
Sure! I'd be happy to help you with this problem step by step.
(a) To differentiate both sides of the equation with respect to x, we need to apply the rules of differentiation to each term.
For the term xy, we need to use the product rule, which states that d/dx(uv) = u * dv/dx + v * du/dx.
In this case, u = x and v = y, so applying the product rule, we get:
d/dx(xy) = x * dy/dx + y * dx/dx = x * dy/dx + y
For the term 7x, we can use the power rule of differentiation, which states that d/dx(x^n) = n * x^(n-1).
In this case, n = 1, so we have:
d/dx(7x) = 7 * d/dx(x^1) = 7 * 1 * x^(1-1) = 7
For the term 5y, we again use the power rule of differentiation with n = 1:
d/dx(5y) = 5 * d/dx(y^1) = 5 * 1 * y^(1-1) = 5
Now, by differentiating the entire equation, we have:
x * dy/dx + y + 7 - 5 = 0
xy' + y + 2 = 0
(b) To solve for y' in terms of x and y, we isolate the term with the derivative:
xy' = -y - 2
Dividing both sides by x, we get:
y' = (-y - 2) / x
(c) To calculate dy/dx at the point (4, -4), we substitute x = 4 and y = -4 into the equation we just found for y':
y' = (-(-4) - 2) / 4
= (4 - 2) / 4
= 2 / 4
= 1/2
So, at the point (4, -4), the value of dy/dx is 1/2.
I hope this explanation helps! Let me know if you have any further questions.