y = (ln x)^(cos4x)

Use logarithmic differentiation to find the derivative of the function.

Well, well, well, logarithmic differentiation, huh? You sure know how to make things interesting! But fear not – the Clown Bot is here to help!

To find the derivative of the function y = (ln x)^(cos 4x), we'll break out our logarithmic differentiation skills. Are you ready? Let's go!

First, we'll take the natural logarithm of both sides of the equation in order to simplify things. So we have:

ln y = ln((ln x)^(cos 4x))

Now, let's apply the power rule of logarithms, which states that ln(a^b) = b * ln(a). So, we can rewrite the equation as:

ln y = (cos 4x) * ln (ln x)

Next, we'll differentiate both sides of the equation with respect to x using the chain rule. On the left side, we need to use implicit differentiation since y is a function of x:

(1/y) * dy/dx = [(d/dx)(cos 4x)] * ln(ln x) + (cos 4x) * [(d/dx)(ln(ln x))]

Seems like we're doing pretty well so far. But here's where things get a bit trickier. We need to find the derivatives of the terms involved. Let me tell you, the derivative of cosine is negative sine, and the derivative of natural logarithm is 1/x. So, we can rewrite the equation as:

(1/y) * dy/dx = -4sin 4x * ln(ln x) + (cos 4x) * (1/ln x) * (1/x)

Phew! That was quite a ride, wasn't it? Now we just need to solve for dy/dx. We start by multiplying both sides by y:

dy/dx = y * [ -4sin 4x * ln(ln x) + (cos 4x) * (1/ln x) * (1/x) ]

Finally, we substitute y back in:

dy/dx = [ (ln x)^(cos 4x) ] * [ -4sin 4x * ln(ln x) + (cos 4x) * (1/ln x) * (1/x) ]

And that, my friend, is your derivative using logarithmic differentiation. I hope you enjoyed the ride! Stay silly and keep rocking those math problems!

To find the derivative of the function y = (ln x)^(cos(4x)) using logarithmic differentiation, follow these steps:

Step 1: Take the natural logarithm of both sides of the equation y = (ln x)^(cos(4x)):
ln(y) = ln((ln x)^(cos(4x)))

Step 2: Apply the power rule of logarithms to simplify the equation:
ln(y) = (cos(4x)) * ln(ln x)

Step 3: Differentiate both sides of the equation with respect to x:
(1/y) * dy/dx = -sin(4x) * ln(ln x) + (cos(4x)) * (1/ln x) * (1/x)

Step 4: Solve for dy/dx by multiplying both sides of the equation by y:
dy/dx = y * (-sin(4x) * ln(ln x) + (cos(4x)) * (1/ln x) * (1/x))

Step 5: Substitute the value of y back into the equation:
dy/dx = (ln x)^(cos(4x)) * (-sin(4x) * ln(ln x) + (cos(4x)) * (1/ln x) * (1/x))

Therefore, the derivative of y = (ln x)^(cos(4x)) is dy/dx = (ln x)^(cos(4x)) * (-sin(4x) * ln(ln x) + (cos(4x)) * (1/ln x) * (1/x)).

To find the derivative of the function y = (ln x)^(cos(4x)), we can use logarithmic differentiation. Here's how we can do it step by step:

1. Take the natural logarithm of both sides of the equation: ln y = ln((ln x)^(cos(4x))).

2. Use the logarithmic property: ln(a^b) = b * ln(a). Applying this to our equation, we get ln y = cos(4x) * ln(ln x).

3. Now we will differentiate both sides of the equation with respect to x. The derivative of the left side is (1/y) * (dy/dx). The derivative of the right side requires the product rule: d/dx (cos(4x) * ln(ln x)) = cos(4x) * d/dx (ln(ln x)) + ln(ln x) * d/dx (cos(4x)).

4. Differentiate the first term on the right side. Using the chain rule, we get d/dx (ln(ln x)) = (1/ln(x)) * (1/x) * dx/dx = (1/(x ln(x))) * (1/x) = 1/(x^2 ln(x)).

5. Differentiate the second term on the right side. Since the derivative of cos(4x) is -4sin(4x), we have d/dx (cos(4x)) = -4sin(4x).

6. Substituting our results back into the equation, we have (1/y) * (dy/dx) = cos(4x) * (1/(x^2 ln(x))) - 4sin(4x) * ln(ln x).

7. Multiply both sides of the equation by y to isolate dy/dx. This gives us dy/dx = y * [cos(4x) * (1/(x^2 ln(x))) - 4sin(4x) * ln(ln x)].

8. Replace y with the original expression (ln x)^(cos(4x)). This gives us the final answer: dy/dx = (ln x)^(cos(4x)) * [cos(4x) * (1/(x^2 ln(x))) - 4sin(4x) * ln(ln x)].

Therefore, the derivative of y = (ln x)^(cos(4x)) is dy/dx = (ln x)^(cos(4x)) * [cos(4x) * (1/(x^2 ln(x))) - 4sin(4x) * ln(ln x)].

lny = ln[(lnx)^(cos(4x))]

= cos(4x) * ln(ln(x))
y'/y = cos(4x)*1/ln(x) * 1/x + ln(ln(x))(-4sin(4x)

y' = (ln x)^(cos4x) [cos(4x)*1/ln(x) * 1/x + ln(ln(x))(-4sin(4x) ]