z=cos(4 theta)
what is the derivative.
I know that the derivative of cos is -sin, but what do i do with the 4 theta. does that just become 4?
dz/d(theta) = -sin(4theta) * 4
= -4sin(4theta)
in general if
y = cos u
then dy/dx = -sin u (du/dx)
To find the derivative of z = cos(4 theta), we can use the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative is given by f'(g(x)) times g'(x).
Let's break it down step-by-step:
Step 1: Identify the outer function and the inner function.
In this case, the outer function is cos and the inner function is 4 theta.
Step 2: Take the derivative of the outer function.
The derivative of cos(x) is -sin(x).
Step 3: Take the derivative of the inner function.
The derivative of 4 theta with respect to theta is simply 4.
Step 4: Combine the derivatives using the chain rule.
Apply the chain rule by multiplying the derivative of the outer function with the derivative of the inner function:
(-sin(4 theta)) * 4
So, the derivative of z = cos(4 theta) is -4 sin(4 theta).
To find the derivative of the function z = cos(4θ), you can apply the chain rule of differentiation.
The chain rule states that if you have a function of the form f(g(x)), the derivative of that function with respect to x is given by the product of the derivative of f with respect to g, and the derivative of g with respect to x.
In this case, f(x) = cos(x), and g(x) = 4θ.
The derivative of f(x) = cos(x) with respect to x is -sin(x).
Now, we need to find the derivative of g(x) = 4θ with respect to θ. Since θ is the variable of differentiation in this case, we treat 4θ as a constant.
The derivative of g(x) = 4θ with respect to θ is 4.
So, applying the chain rule, the derivative of z = cos(4θ) with respect to θ is given by:
dz/dθ = (d(cos(4θ))/d(4θ)) * (d(4θ)/dθ).
Since (d(cos(4θ))/d(4θ)) = -sin(4θ), and (d(4θ)/dθ) = 4, the derivative becomes:
dz/dθ = -sin(4θ) * 4.
Hence, the derivative of z = cos(4θ) is dz/dθ = -4sin(4θ).