Differentiate: y=sin(2z)+cos^2z
dy/dz = 2 cos (2z) -2 sin (2z)(cos (2z)
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To differentiate the function y = sin(2z) + cos^2(z), we can use the rules of differentiation. Let's differentiate each term of the function separately and then combine the results.
1. Differentiating sin(2z):
The derivative of sin(2z) with respect to z can be found by applying the chain rule. The chain rule states that if we have a function of the form f(g(z)), where f(u) is a function of u and g(z) is a function of z, then the derivative with respect to z is given by f'(g(z)) * g'(z).
In this case, f(u) = sin(u) and g(z) = 2z. Taking the derivative of sin(u) with respect to u, we get cos(u). Taking the derivative of g(z) with respect to z, we get 2.
So, the derivative of sin(2z) with respect to z is cos(2z) * 2, which simplifies to 2cos(2z).
2. Differentiating cos^2(z):
The derivative of cos^2(z) with respect to z can be found using the chain rule again. Let's rewrite cos^2(z) as (cos(z))^2.
In this case, f(u) = u^2 and g(z) = cos(z). Taking the derivative of u^2 with respect to u, we get 2u. Taking the derivative of g(z) with respect to z, we get -sin(z) (since the derivative of cos(z) is -sin(z)).
So, the derivative of (cos(z))^2 with respect to z is 2cos(z) * -sin(z), which simplifies to -2cos(z)sin(z).
Now, let's combine the derivatives of the two terms:
dy/dz = d/dz[sin(2z)] + d/dz[cos^2(z)]
= 2cos(2z) + (-2cos(z)sin(z))
= 2cos(2z) - 2cos(z)sin(z)
Therefore, the derivative of y = sin(2z) + cos^2(z) with respect to z is 2cos(2z) - 2cos(z)sin(z).