Differentiate y=cos^5(4x)
Using chain rule,
dy/dx = 5cos^4(4x) * (-sin(4x)) * 4
= -80cos^4(4x)sin(4x)
To differentiate the given function, y = cos^5(4x), we will use the chain rule. Let's proceed step-by-step:
Step 1: Start by expressing the function as the composition of two functions:
f(u) = u^5, where u = cos(4x)
Step 2: Differentiate the inner function (4x) with respect to x using the chain rule:
d(u)/d(x) = d(cos(4x))/d(x)
To differentiate cosine, we use the chain rule: d(cos(f(x)))/d(x) = -sin(f(x)) * f'(x)
In this case, f(x) = 4x, so the derivative is:
d(cos(4x))/d(x) = -sin(4x) * d(4x)/d(x)
Since d(4x)/d(x) = 4, this simplifies to:
d(u)/d(x) = -4sin(4x)
Step 3: Substitute the derived expression back into the original function:
dy/dx = d(f(u))/d(u) * d(u)/d(x)
= d(u^5)/d(u) * -4sin(4x)
= 5u^(5-1) * -4sin(4x)
= -20u^4 * sin(4x)
Step 4: Replace the u with the original expression, which is cos(4x):
dy/dx = -20 * cos^4(4x) * sin(4x)
Therefore, the derivative of y = cos^5(4x) with respect to x is -20 * cos^4(4x) * sin(4x).