Use implicit differentiation to find the derivative of the family of curves
sin(xy)+x5+y5=c.
dy/dx=?
I got
(-5x^4-ycos(xy))/(xcos(xy)+5y^4)). what am I doing wrong? Help please
sin(xy)+x^5+y^5=c
cos(xy)(y+xy') + 5x^4 + 5y^4 y' = 0
y*cos(xy) + x*cos(xy)y' + 5x^4 + 5y^4 y' = 0
(x*cos(xy) + 5y^4)y' = -(y*cos(xy) + 5x^4)
y' =
y*cos(xy) + 5x^4
- -------------------------
x*cos(xy) + 5y^4
Well, it seems like you're on the right track, but there might be a couple of mistakes in your calculations. Let's go through it step by step.
We have the equation sin(xy) + x^5 + y^5 = c. To find the derivative dy/dx, we'll treat y as an implicit function of x and differentiate both sides of the equation with respect to x.
Let's start by differentiating sin(xy):
d/dx(sin(xy)) = cos(xy) * (y + xy') ... (1)
Next, let's differentiate x^5 with respect to x:
d/dx(x^5) = 5x^4 ... (2)
And for y^5, we have:
d/dx(y^5) = 5y^4 * y' ... (3)
Now, let's differentiate the constant term c, which is zero:
d/dx(c) = 0 ... (4)
Combining equations (1), (2), (3), and (4), we get:
cos(xy) * (y + xy') + 5x^4 + 5y^4 * y' = 0
Now, let's isolate dy/dx:
cos(xy) * y + x * cos(xy) * y' + 5x^4 + 5y^4 * y' = 0
cos(xy) * y + 5x^4 = -x * cos(xy) * y' - 5y^4 * y'
(cos(xy) * y + 5x^4) / (-x * cos(xy) - 5y^4) = y'
So, the derivative dy/dx is given by:
dy/dx = (cos(xy) * y + 5x^4) / (-x * cos(xy) - 5y^4)
Keep in mind that this is just the derivative of the implicit function y with respect to x, so it won't be in terms of x and y separately.
I hope this helps!
To find the derivative of the given function using implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x, treating y as a function of x.
d/dx(sin(xy) + x^5 + y^5) = d/dx(c)
Step 2: Apply the chain rule to the terms involving y. For the term sin(xy) and y^5, we will use the chain rule.
[d/dx(sin(xy))] + [d/dx(x^5)] + [d/dx(y^5)] = 0
Step 3: Differentiate each term separately.
For the term sin(xy), we apply the chain rule. The derivative of sin(u) with respect to x is cos(u) times the derivative of u with respect to x.
Chain rule: d/dx sin(u) = cos(u) * (du/dx)
Using this rule for our term sin(xy), we have:
cos(xy) * (d/dx(xy))
For the term x^5, the derivative with respect to x is:
5x^4
For the term y^5, we apply the chain rule similarly to the previous term:
5y^4 * (d/dx(y))
Step 4: Combine all the derivatives and simplify the expression:
cos(xy) * (d/dx(xy)) + 5x^4 + 5y^4 * (dy/dx) = 0
Step 5: Solve for dy/dx:
Rearrange the equation to isolate dy/dx:
cos(xy) * (d/dx(xy)) + 5y^4 * (dy/dx) = -5x^4
Now, isolate dy/dx:
5y^4 * (dy/dx) = -5x^4 - cos(xy) * (d/dx(xy))
Divide through by 5y^4:
dy/dx = (-5x^4 - cos(xy) * (d/dx(xy))) / (5y^4)
To simplify further, we need to find the derivative of xy with respect to x.
For the term xy, we use the product rule:
(d/dx(xy)) = x * (d/dx(y)) + y * (d/dx(x))
Using the product rule, we have:
x * (dy/dx) + y * (1)
Substituting this back into the equation for dy/dx, we get:
dy/dx = (-5x^4 - cos(xy) * (x * (dy/dx) + y)) / (5y^4)
Now, we have an equation for dy/dx in terms of x, y, and dy/dx.
To find the derivative of the given curve using implicit differentiation, follow these steps:
Step 1: Differentiate both sides of the equation with respect to x. Treat y as a function of x and apply the chain rule.
sin(xy) + x^5 + y^5 = c
Differentiating both sides with respect to x:
d/dx [sin(xy)] + d/dx [x^5] + d/dx [y^5] = d/dx [c]
Step 2: Apply the product rule and chain rule as needed:
For the term sin(xy), use the chain rule:
d/dx [sin(xy)] = cos(xy) * (y + x * dy/dx)
For the term x^5, differentiate normally:
d/dx [x^5] = 5x^4
For the term y^5, use the chain rule:
d/dx [y^5] = 5y^4 * dy/dx
Since c is a constant, the derivative is zero:
d/dx [c] = 0
Step 3: Substitute these values back into the equation:
cos(xy) * (y + x * dy/dx) + 5x^4 + 5y^4 * dy/dx = 0
Step 4: Solve for dy/dx by isolating the term:
Group the terms with dy/dx on one side of the equation:
cos(xy) * x * dy/dx + 5y^4 * dy/dx = -cos(xy) * y - 5x^4
Factor out dy/dx:
dy/dx * (cos(xy) * x + 5y^4) = -cos(xy) * y - 5x^4
Divide both sides by (cos(xy) * x + 5y^4):
dy/dx = (-cos(xy) * y - 5x^4) / (cos(xy) * x + 5y^4)
So, the correct derivative should be:
dy/dx = (-cos(xy) * y - 5x^4) / (cos(xy) * x + 5y^4)
It appears that you made an error in the denominator, as you wrote (xcos(xy) + 5y^4), but it should be (cos(xy) * x + 5y^4) as shown above.