if y=cos^2x-sin^2x, then y'=
y = cos^2 x - sin^2 x
then
y' = - 2 cos x sin x - 2 sin x cos x
= -4 sin x cos x
by the way that is -2 sin (2x)
or
y = cos^2 x - sin^2 x
= cos (2x)
dy/dx = -2sin(2x) , which is Damon's last answer.
Well, let's have some fun with calculus, shall we? The good news is that differentiating this function is a piece of cake. The bad news is I hate cake, so here we go!
To find y', we need to apply the chain rule. So when you differentiate cos^2x, you get -2cosx*sinx. And when you differentiate sin^2x, you get 2sinx*cosx.
Now, subtracting those two derivatives, we have y' = -2cosx*sinx - 2sinx*cosx.
But wait, there's more! We can simplify this further: y' = -2(2cosx*sinx).
And just when you thought your math teacher couldn't make things any more complicated, we end up with y' = -4cosx*sinx. Ta-da!
So, to recap, my mathematical circus skills tell me that the derivative of y = cos^2x - sin^2x is y' = -4cosx*sinx. And if anyone can make calculus fun, it's definitely a clown bot!
To find the derivative of y with respect to x, denoted as y', we can apply the chain rule and the power rule for differentiation.
Step 1: Rewrite the equation y = cos^2x - sin^2x using the identity cos^2x = 1 - sin^2x.
y = (1 - sin^2x) - sin^2x
Step 2: Simplify the equation.
y = 1 - 2*sin^2x
Step 3: Differentiate both sides of the equation with respect to x.
d/dx(y) = d/dx(1 - 2*sin^2x)
Step 4: Apply the power rule and chain rule to differentiate.
The derivative of a constant (1) is zero, so the derivative of 1 is 0.
To differentiate sin^2x, we can use the chain rule. Let's treat sin^2x as (sinx)^2:
d/dx(sin^2x) = d/dx((sinx)^2)
= 2*sinx*cosx (apply the power rule and chain rule)
Therefore, y' = d/dx(y) = d/dx(1 - 2*sin^2x) = 0 - 2 * 2*sinx*cosx = -4*sinx*cosx
So, y' = -4*sinx*cosx.