what is the derivative of:
y=2sinxcosx
to ways to do it
1. recognize that 2sinxcosx = sin(2x)
then dy/dx = 2cos(2x)
or
2. y = 2sinxcosx
dy/dx = 2sinx(-sinx) + 2cosx(cosx)
= 2(cos^2x - sin^2x)
which just happens to be 2cos(2x) as above
To find the derivative of y = 2sin(x)cos(x), we can use the product rule. The product rule states that if we have a function f(x) multiplied by g(x), the derivative of their product is given by:
(f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
Let's find the derivative step-by-step:
Step 1: Identify the functions f(x) and g(x).
In this case, f(x) = 2sin(x) and g(x) = cos(x).
Step 2: Find the derivative of f(x), denoted as f'(x).
The derivative of sin(x) is cos(x). Therefore, d/dx (2sin(x)) = 2cos(x).
Step 3: Find the derivative of g(x), denoted as g'(x).
The derivative of cos(x) is -sin(x). Therefore, d/dx (cos(x)) = -sin(x).
Step 4: Apply the product rule to find the derivative of y.
Using the product rule, we have:
d/dx (2sin(x)cos(x)) = (2cos(x))(cos(x)) + (2sin(x))(-sin(x))
= 2cos^2(x) - 2sin^2(x)
Therefore, the derivative of y = 2sin(x)cos(x) is dy/dx = 2cos^2(x) - 2sin^2(x).
To find the derivative of y=2sin(x)cos(x), we can use the product rule.
The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product, y(x) = u(x)v(x), can be found using the formula:
dy/dx = u(x) * dv/dx + v(x) * du/dx
In this case, u(x) = 2sin(x) and v(x) = cos(x).
To find du/dx, we need to take the derivative of u(x) with respect to x. In this case, du/dx will be the derivative of sin(x) with respect to x times 2. The derivative of sin(x) is cos(x), so du/dx can be written as 2cos(x).
Similarly, to find dv/dx, we need to take the derivative of v(x) with respect to x. In this case, dv/dx will be the derivative of cos(x) with respect to x. The derivative of cos(x) is -sin(x), so dv/dx can be written as -sin(x).
Now we can substitute these values back into the formula for the product rule:
dy/dx = (2sin(x)) * (-sin(x)) + (cos(x)) * (2cos(x))
Simplifying this expression will give us the derivative of y with respect to x.