Find the 1st derivative of the function:
y = 1/4 sin^2 (2x)
So confused...
dy/dx
=(1/4)d(sin^2(2x))/d(sin(2x))*d(sin(2x))/dx
=(1/4)*2sin(2x)*cos(2x)*2
=sin(2x)cos(2x)
=(1/2)sin(4x)
To find the first derivative of the function y = 1/4 sin^2 (2x), we will apply the chain rule and the power rule.
The chain rule states that for a composite function f(g(x)), the derivative of f with respect to x is given by f'(g(x)) * g'(x).
First, let's find the derivative of the function inside the parentheses.
Let u = 2x.
Now, we have y = 1/4 sin^2 (u).
To find the derivative of this function, we differentiate it as follows:
dy/du = d/dx (1/4 sin^2 (u))
To differentiate sin^2 (u), we will apply the chain rule.
Using the power rule, the derivative of sin^2 (u) is 2sin(u) * cos(u).
So, dy/du = (1/4) * 2sin(u) * cos(u)
dy/du = 1/2 sin(u) * cos(u)
Now, we need to solve for dy/dx, not dy/du. We can do that using the chain rule.
dy/dx = (dy/du) * (du/dx)
Since u = 2x, du/dx = 2.
Therefore, dy/dx = (1/2 sin(u) * cos(u)) * 2
dy/dx = sin(u) * cos(u)
Substituting back u = 2x, we get:
dy/dx = sin(2x) * cos(2x)
So, the 1st derivative of the function y = 1/4 sin^2 (2x) is dy/dx = sin(2x) * cos(2x).
To find the first derivative of the function y = (1/4)sin^2(2x), we can use the power rule and the chain rule for differentiation.
Step 1: Rewrite the function using the power rule.
We can rewrite the function as y = (1/4)(sin(2x))^2.
Step 2: Apply the power rule.
To differentiate y with respect to x, we need to apply the power rule, which states that if we have a function of the form y = u^n, where u is a function of x and n is a constant, then the derivative of y with respect to x is given by dy/dx = n*u^(n-1)*du/dx.
In this case, u = sin(2x) and n = 2. So, let's differentiate the function.
dy/dx = (2)(sin(2x))^(2-1) * d(sin(2x))/dx.
Step 3: Apply the chain rule.
The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative of f(g(x)) with respect to x is given by df/dx = f'(g(x)) * g'(x).
In this case, f(u) = u^2 and g(x) = sin(2x). So, let's find the derivatives of f(u) and g(x) separately.
df/du = 2u
dg/dx = d(sin(2x))/dx
Now, let's substitute these derivatives into our equation.
dy/dx = (2)(sin(2x))^(2-1) * d(sin(2x))/dx
= 2(sin(2x)) * d(sin(2x))/dx.
Step 4: Differentiate sin(2x).
To differentiate sin(2x) with respect to x, we can use the chain rule again.
d(sin(2x))/dx = cos(2x) * d(2x)/dx
= 2cos(2x).
Now, substitute this back into our equation.
dy/dx = 2(sin(2x)) * d(sin(2x))/dx
= 2(sin(2x)) * 2cos(2x).
Simplifying further:
dy/dx = 4sin(2x)cos(2x).
And there you have it! The first derivative of the function y = (1/4)sin^2(2x) is dy/dx = 4sin(2x)cos(2x).