r=log(cos^2(4theta)) what is the derivative of this?
just use the chain rule:
let r = log(u)
dr/dθ = 1/u du/dθ
let u = cos^2(v)
du/dθ = 2cosv(-sinv) dv/dθ
let v = 4θ
dv/dθ = 4
So, dr/dθ = sec^2(4θ) (-2cos4θsin4θ) (4)
= -8tan4θ
Or, you could just recall that
log(cos^2(4θ)) = 2log cos 4θ
so a simpler application of the chain rule would be
2/cos4θ (-4sin4θ)
= -8tan4θ
To find the derivative of r = log(cos^2(4θ)), we can use the chain rule. Let's break it down step-by-step.
Step 1: Start with the given function r = log(cos^2(4θ)).
Step 2: Rewrite the function using exponential form: r = ln(cos^2(4θ)).
Step 3: Apply the chain rule. The chain rule states that if we have a composition of functions inside a logarithm, we need to differentiate the inner and outer functions separately.
Step 4: Differentiate the outer function, which is the natural logarithm (ln): d/dθ(ln(u)) = 1/u * du/dθ, where u = cos^2(4θ).
Step 5: Differentiate the inner function, which is cos^2(4θ). To differentiate a composition of functions, we need to apply the chain rule again.
Step 6: Let's differentiate cos^2(4θ) step-by-step.
- The derivative of cos^2(x) is -sin(2x) * 2 (using the chain rule).
- Apply this rule to our function by multiplying by the derivative of the exponent (4θ): d/dθ(cos^2(4θ)) = -sin(2(4θ)) * 2 * 4.
Step 7: Simplify the expression from Step 6: d/dθ(cos^2(4θ)) = -8sin(8θ).
Step 8: Substitute the result from Step 7 into Step 4: d/dθ(ln(u)) = 1/u * du/dθ = 1/cos^2(4θ) * (-8sin(8θ)).
Step 9: Simplify the expression: d/dθ(ln(u)) = -8sin(8θ)/cos^2(4θ).
So, the derivative of r = log(cos^2(4θ)) is -8sin(8θ)/cos^2(4θ).
To find the derivative of the function r = log(cos^2(4theta)), we will need to use the chain rule and the derivative of the natural logarithm function.
Let's break it down step by step:
Step 1: Rewrite the function
r = log(cos^2(4theta))
r = 2log(cos(4theta))
Step 2: Apply the chain rule
To differentiate the function, we'll need to differentiate the outer function (log) and the inner function (cos(4theta)) separately.
Derivative of the outer function: d/dx (log(u)) = 1/u * du/dx
So, applying this to r = 2log(cos(4theta)), the derivative of the outer function is:
dr/du = 2 * 1/u
Derivative of the inner function: d/dx (cos(4theta)) = -sin(4theta) * d/dx(4theta)
Since d/dx(4theta) is simply 4, the derivative of the inner function is:
du/dx = -4sin(4theta)
Step 3: Combine the results
By applying the chain rule, we multiply the derivative of the outer function by the derivative of the inner function:
dr/dx = dr/du * du/dx
dr/dx = 2 * 1/u * -4sin(4theta)
Step 4: Substitute u back in
Since u is equal to cos(4theta), we can substitute it back in:
dr/dx = 2 * 1/cos(4theta) * -4sin(4theta)
Simplifying further:
dr/dx = -8sin(4theta) / cos(4theta)
Thus, the derivative of the function r = log(cos^2(4theta)) is -8sin(4theta) / cos(4theta).