A mathematical themed illustration showcasing the process of finding derivatives in calculus. In the center, there's an abstract representation of a cosine wave - it could be visualized as a smooth, wavy line that undulates symmetrically along an imaginary horizontal axis. Complementing this, create a depiction of squaring operation (symbolic, not literal) to reflect cos^2. To signify '4x', show four copies of this pattern intersecting. Make sure to incorporate themes of transformation and change often associated with the concept of derivatives. The image contains no text.

Find the derivative of

cos^2(4x)

Use the chain rule:

d(f(u(x))/dx=df/du.du/dx
d(cos²(4x))/dx
=2cos(4x)d(cos(4x))/dx
=2cos(4x)(-sin(4x))d(4x)/dx
=2cos(4x)(-sin(4x))(4)
=-8sin(4x)cos(4x)
=-4sin(8x)

The last simplification may not be required or recognized by a computerized answer.

Well, I hate to break it to you, but finding derivatives is no joke. The derivative of cos^2(4x) requires a little bit of math, but I'll try my best to explain it in a fun way.

Let's start by using the chain rule. Think of it like an intricate juggling routine. The chain rule says that if we have a function within a function, we need to differentiate both functions and then multiply them together.

So, the first thing we need to do is differentiate the function cos^2(4x). Hold onto your funny hat because here comes the math: when you differentiate cos^2(4x), you get -8cos(4x)sin(4x).

Let's break it down: the derivative of cos^2(4x) is equal to -8cos(4x)sin(4x). It's like juggling negative eights, cosine four x's, and sine four x's all at the same time. Pretty impressive, huh?

So, the derivative of cos^2(4x) is -8cos(4x)sin(4x). I hope that brought a smile to your face, or at least made you chuckle a bit. Keep on laughing and learning!

To find the derivative of cos^2(4x), we can apply the chain rule. Let's break it down step-by-step:

Step 1: Identify the function within the function.
In this case, the inner function is 4x.

Step 2: Find the derivative of the outer function.
The derivative of cos^2(x) can be found using the chain rule:
d/dx [cos^2(x)] = 2cos(x) * (-sin(x))

Step 3: Multiply the derivative of the outer function by the derivative of the inner function.
Since the inner function is 4x, its derivative with respect to x is simply 4.

Step 4: Put it all together.
Applying the chain rule, we have:
d/dx [cos^2(4x)] = 2cos(4x) * (-sin(4x)) * 4

Simplifying this expression gives the final result:
d/dx [cos^2(4x)] = -8cos(4x)sin(4x)

To find the derivative of cos^2(4x), you can use the chain rule. The chain rule states that if you have a composite function, f(g(x)), then the derivative of the composite function is given by the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function.

Let's break down the function cos^2(4x) into its components:
- The outer function is cos^2(u), where u = 4x.
- The inner function is u = 4x.

We can find the derivatives of the outer and inner functions separately.

1. Derivative of the outer function:
To find the derivative of cos^2(u), we treat it as (cos(u))^2 and apply the chain rule.
Let y = cos(u).
Therefore, y^2 = (cos(u))^2.
Now, let's find the derivative of y^2 with respect to y, which is 2y.
Therefore, the derivative of cos^2(u) with respect to u is 2cos(u).

2. Derivative of the inner function:
To find the derivative of u = 4x, we treat it as a simple linear equation.
The derivative of 4x with respect to x is 4.

Now, we can apply the chain rule by multiplying the derivatives of the outer and inner functions:
The derivative of cos^2(4x) = (2cos(u)) * (4).

Substituting u = 4x back in, we get:
The derivative of cos^2(4x) = (2cos(4x)) * (4).

Therefore, the final derivative of cos^2(4x) is 8cos(4x).