find the derivatives
1. y=sin^3(2x)(cos2x)
To find the derivative of the given function, which is y = sin^3(2x)cos(2x), we can use the product rule and the chain rule. Here's how to do it step by step:
Step 1: Rewrite the function using the power rule for sine:
y = (sin(2x))^3 * cos(2x)
Step 2: Apply the product rule. Let's call the first term u and the second term v:
u = (sin(2x))^3
v = cos(2x)
Step 3: Find the derivatives of u and v:
To find the derivative of u, we need to use the chain rule. Let's call the inner function g(x) = sin(2x) and the outer function f(x) = x^3. The derivative of u, du/dx, is given by:
du/dx = 3(g(x))^2 * g'(x)
To calculate g'(x), we need to differentiate g(x), which is sin(2x). Using the chain rule, we have:
g'(x) = cos(2x) * 2
So, du/dx = 3(sin(2x))^2 * cos(2x) * 2
To find the derivative of v, which is cos(2x), we differentiate it directly:
dv/dx = -sin(2x)
Step 4: Apply the product rule formula to find the derivative of y:
dy/dx = u * dv/dx + v * du/dx
Substituting the values of u, v, du/dx, and dv/dx, we have:
dy/dx = (sin^3(2x))( -sin(2x)) + (cos(2x))(3(sin^2(2x))(cos(2x))(2))
Step 5: Simplify the expression:
dy/dx = -sin^4(2x) - 6(sin^2(2x))(cos^2(2x))
The derivative of y with respect to x is given by dy/dx = -sin^4(2x) - 6sin^2(2x)cos^2(2x).
And that's how you find the derivative of y = sin^3(2x)cos(2x)!