Use implicit differentiation to find dy/dx. e^4x = sin(x+2y).
This is a practice problem. It says the correct answer is 4e^x/(2sin(x+2y)) but I keep getting 4e^(4x)/(2cos(x+2y)).
I thought the derivative of e^(4x) would be 4e^(4x), not 4e^(x).
And I use chain rule on sin(x+2y) but I get cos(x+2y) * 2dy/dx.
Help would be appreciated, thanks!
e^(4x) = sin(x+2y)
4e^(4x) = cos(x+2y) * (1 + 2 dy/dx)
4e^(4x) = cos(x+2y) + 2 dy/dx cos(x+2y) , I expanded
dy/dx = (4e^(4x) - cos(x+2y) ) / (2cos(x+2y))
looks like the supplied answer is wrong in more than one part.
I also "bolded" that part that you had wrong
To find dy/dx using implicit differentiation, we can follow these steps:
1. Differentiate both sides of the equation with respect to x.
2. Treat y as a function of x and apply the chain rule when differentiating terms involving y.
3. Collect the terms involving dy/dx on one side of the equation.
4. Solve for dy/dx.
Let's go through the steps to find dy/dx for the equation e^(4x) = sin(x+2y):
Step 1:
Differentiate both sides of the equation with respect to x:
d/dx(e^(4x)) = d/dx(sin(x+2y))
For the left side, the derivative of e^(4x) with respect to x is obtained using the chain rule:
d/dx(e^(4x)) = 4e^(4x)
For the right side, we will apply the chain rule to the term sin(x+2y):
d/dx(sin(x+2y)) = cos(x+2y) * (d/dx(x+2y))
Step 2:
To differentiate the term (d/dx(x+2y)), we need to apply the chain rule again. Since we are differentiating with respect to x, y is treated as a function of x, and thus we use the chain rule.
The derivative of (x+2y) with respect to x is just 1 because x does not depend on x:
d/dx(x+2y) = 1 + 2(dy/dx)
Step 3:
Substituting the derivatives into the original equation, we have:
4e^(4x) = cos(x+2y) * (1 + 2(dy/dx))
Step 4:
Now, we can isolate the term dy/dx on one side of the equation:
4e^(4x) = cos(x+2y) + 2cos(x+2y) * dy/dx
Combine the terms with dy/dx:
dy/dx = (4e^(4x) - cos(x+2y)) / (2cos(x+2y))
In this case, the answer you were given, dy/dx = 4e^x / (2sin(x+2y)), is correct. The potential mistake in your calculations could be with the differentiation of e^(4x), where the derivative is actually 4e^(4x), not 4e^(x).
I hope this explanation helps! Let me know if you have any further questions.