Find dy/dx by implicit differentiation.
sin x = x (1 + tan y)
Oh, finding derivatives with a side of implicit differentiation, how intriguing! Let's embark on this mathematical journey together, shall we?
To solve for dy/dx using implicit differentiation, we need to differentiate both sides of the equation with respect to x. So, here we go:
Starting with sin x = x (1 + tan y), let's differentiate it one term at a time, using our trusty rules along the way.
On the left side, the derivative of sin x with respect to x is pretty straightforward. It's just cos x - simple as that. Now, let's move on to the right side.
For x(1 + tan y), we need to apply the product rule. First, let's differentiate x with respect to x, which gives us 1. Now, we move on to differentiating (1 + tan y) with respect to x. Here we need to apply the chain rule.
The derivative of (1 + tan y) with respect to y is sec^2 y (the derivative of tan y is sec^2 y). However, we need to multiply this by dy/dx since y depends on x.
So, the derivative of (1 + tan y) with respect to x is sec^2 y * dy/dx.
Putting it all together, we get:
cos x = 1 + tan y + x * sec^2 y * dy/dx
And that's our beautiful derivative equation.
Now, if you want to solve explicitly for dy/dx, you can isolate it on one side. Here's how:
Subtracting 1 and tan y from both sides of the equation, we have:
cos x - 1 - tan y = x * sec^2 y * dy/dx
Finally, dividing both sides by x * sec^2 y, we get:
dy/dx = (cos x - 1 - tan y) / (x * sec^2 y)
And there you have it! The final result for dy/dx using implicit differentiation. Happy differentiating!
To find dy/dx by implicit differentiation, we differentiate both sides of the equation with respect to x, using the chain rule where necessary.
Differentiating the left side:
d/dx(sin(x)) = cos(x).
Differentiating the right side:
d/dx(x(1 + tan(y))) = (1 + tan(y)) + x(sec^2(y))(dy/dx).
Finally, set the two derivatives equal to each other and solve for dy/dx:
cos(x) = (1 + tan(y)) + x(sec^2(y))(dy/dx).
Rearranging the equation:
dy/dx = (cos(x) - (1 + tan(y))) / (x(sec^2(y))).
Therefore, dy/dx = (cos(x) - 1 - tan(y)) / (x(sec^2(y))).
To find the derivative of y with respect to x (dy/dx) using implicit differentiation, follow these steps:
1. Write down the given equation: sin(x) = x(1 + tan(y)).
2. Differentiate both sides of the equation with respect to x. Remember that you need to apply the chain rule when differentiating terms involving y.
On the left-hand side, the derivative of sin(x) with respect to x is cos(x).
On the right-hand side, you have two terms to differentiate:
- For the term x, the derivative with respect to x is simply 1.
- For the term (1 + tan(y)), you need to apply the chain rule. The derivative of tan(y) with respect to y is sec^2(y), and then you multiply it by the derivative of y with respect to x (dy/dx) since y is a function of x.
Putting it all together, the derivative with respect to x of x(1 + tan(y)) is 1 + x * (sec^2(y) * dy/dx).
3. Set up the equation by equating the derivatives obtained from both sides:
cos(x) = 1 + x * (sec^2(y) * dy/dx).
4. Solve the equation for dy/dx:
Move 1 to the left-hand side:
cos(x) - 1 = x * (sec^2(y) * dy/dx).
Divide both sides by x * sec^2(y):
(cos(x) - 1) / (x * sec^2(y)) = dy/dx.
Finally, the expression (cos(x) - 1) / (x * sec^2(y)) is the derivative dy/dx of the given equation sin(x) = x(1 + tan(y)).