Find dy/dx for x2 + y2 = 2xy.
2. If 5x2 + y4 = -9 then evaluate d^2y/dx^2 when x = 2 and y = 1. Round your answer to two decimal places. Use the hyphen symbol, -, for negative values.
3. find dy/dx if f(x)=(x+1)^2x
4. Find the slope of the graph of the relation x2y + 4y = 8 at the point (2, 1).
x^2 + y^2 = 2 x y
2 x + 2 y dy/dx = 2 x dy/dx + 2 y
dy/dx (y-x) = y-x
dy/dx = 1
easier way without using calculus:
x^2-2xy+y^2 = 0
(x-y)(x-y) = 0
x = y
slope = 1
Find the slope of the graph of the relation x2y + 4y = 8 at the point (2, 1)
(x^2+4)y = 8
(x^2+4)dy/dx + y (2x) = 0
8 dy/dx + 4 = 0
dy/dx = -1/2
1. To find dy/dx for the equation x^2 + y^2 = 2xy, we can use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(x^2) + d/dx(y^2) = d/dx(2xy)
Step 2: Apply the power rule to differentiate x^2 and the chain rule to differentiate y^2.
2x + 2y(dy/dx) = 2x(dy/dx) + 2y
Step 3: Combine like terms and isolate dy/dx.
2x - 2x(dy/dx) = 2y - 2y(dy/dx)
2x - 2y = (2y - 2x)(dy/dx)
(2x - 2y)/(2y - 2x) = dy/dx
Simplifying further,
(x - y)/(y - x) = dy/dx
Since the denominator is negative, we can multiply both the numerator and denominator by -1 to get the final result:
(dy/dx) = (y - x)/(x - y)
2. To evaluate d^2y/dx^2 when x = 2 and y = 1 for the equation 5x^2 + y^4 = -9, we need to take the second derivative.
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(5x^2) + d/dx(y^4) = d/dx(-9)
Step 2: Apply the power rule to differentiate x^2 and the chain rule to differentiate y^4.
10x + 4y^3(dy/dx) = 0
Step 3: Differentiate again using implicit differentiation.
d/dx(10x) + d/dx(4y^3(dy/dx)) = d/dx(0)
10 + 4(dy/dx) * [3y^2(dy/dx) + 4y^3(d^2y/dx^2)] = 0
Step 4: Substitute x = 2 and y = 1 into the equation.
10 + 4(dy/dx) * [3(1)^2(dy/dx) + 4(1)^3(d^2y/dx^2)] = 0
Step 5: Solve for d^2y/dx^2.
Simplifying further,
10 + 4(dy/dx) * [3(dy/dx) + 4(d^2y/dx^2)] = 0
Substituting x = 2, y = 1,
10 + 4(dy/dx) * [3(dy/dx) + 4(d^2y/dx^2)] = 0
Solving for (d^2y/dx^2),
d^2y/dx^2 = -10/[4(3(1)^2(dy/dx) + 4(1)^3(dy/dx))]
Plug in the value of dy/dx obtained from the previous question to solve for d^2y/dx^2.
3. To find dy/dx for the function f(x) = (x+1)^2x, we can use the power rule and the chain rule.
Step 1: Expand the function.
f(x) = (x+1)^2x = x^3 + 2x^2 + x
Step 2: Differentiate the function using the power rule and the chain rule.
dy/dx = 3x^2 + 4x + 1
4. To find the slope of the graph of the relation x^2y + 4y = 8 at the point (2, 1), we can use implicit differentiation.
Step 1: Differentiate both sides of the equation with respect to x.
d/dx(x^2y) + d/dx(4y) = d/dx(8)
Step 2: Apply the product rule to differentiate x^2y and the power rule to differentiate y.
2xy + x^2(dy/dx) + 4(dy/dx) = 0
Step 3: Substitute x = 2 and y = 1 into the equation.
2(2)(1) + (2)^2(dy/dx) + 4(dy/dx) = 0
Step 4: Solve for dy/dx.
Simplifying further,
4 + 4(dy/dx) + 4(dy/dx) = 0
8(dy/dx) = -4
(dy/dx) = -4/8
(dy/dx) = -1/2
Therefore, the slope of the graph at the point (2, 1) is -1/2.
f(x)=(x+1)^2x
d/dx u^v = v u^(v-1)du/dx + u^v lnu dv/dx
so
f'=2x(x+1)^(2x-1) + 2(x+1)^2x ln(x+1)
If 5x2 + y4 = -9 then evaluate d^2y/dx^2 when x = 2 and y = 1
5 x^2 + y^4 = -9
10 x + 4 y^3 dy/dx = 0
so
dy/dx = -2.5 x/y^3
d^2y/dx^2=-2.5[ y^3 - 3 x y^2 dy/dx]/y^6
at (2,1)
dy/dx = -2.5(2/1) = -5
so
d^2y/dx^2=-2.5[1-6(-5)]/1
= -2.5 [ 31]
= - 77.5