Find the derivative of
y=sin(x)*(sinx+cosx)
just use the product rule
If y = uv then
y' = u'v + uv'
where u=sinx and v=sinx+cosx
y' = cosx(sinx+cosx) + sinx(cosx-sinx)
= sin2x + cos2x
To find the derivative of the function y = sin(x) * (sin(x) + cos(x)), we can use the product rule. The product rule states that if we have a function f(x) multiplied by another function g(x), then the derivative of the product is given by f'(x) * g(x) + f(x) * g'(x).
In this case, let's consider f(x) = sin(x) and g(x) = sin(x) + cos(x). We will first find the derivatives of f(x) and g(x) individually, and then apply the product rule.
1. Derivative of f(x) = sin(x):
The derivative of sin(x) is cos(x). So, f'(x) = cos(x).
2. Derivative of g(x) = sin(x) + cos(x):
The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). Adding them together, we get g'(x) = cos(x) - sin(x).
Now, we can apply the product rule:
y' = f'(x) * g(x) + f(x) * g'(x)
= cos(x) * (sin(x) + cos(x)) + sin(x) * (cos(x) - sin(x))
Expanding the terms:
y' = cos(x) * sin(x) + cos(x) * cos(x) + sin(x) * cos(x) - sin(x) * sin(x)
Simplifying further:
y' = cos(x) * sin(x) + cos^2(x) + sin(x) * cos(x) - sin^2(x)
Finally, rearranging the terms:
y' = cos(x) * sin(x) + cos^2(x) + sin(x) * cos(x) - sin^2(x)
Therefore, the derivative of y = sin(x) * (sin(x) + cos(x)) is y' = cos(x) * sin(x) + cos^2(x) + sin(x) * cos(x) - sin^2(x).