https://file.aiquickdraw.com/mathbot/file/1725904748304h0046kt9.jpg

The question in the image asks you to find \(\frac{dy}{dx} + 5y^2\) when \(y = \cot 5x\).

Let's break it down step by step:

### Step 1: Differentiate \(y = \cot(5x)\) with respect to \(x\)

We use the chain rule for differentiation:
\[
\frac{d}{dx}[\cot(5x)] = -\csc^2(5x) \cdot 5 = -5\csc^2(5x)
\]
So, 
\[
\frac{dy}{dx} = -5\csc^2(5x)
\]

### Step 2: Substituting into the given expression

We are asked to find \(\frac{dy}{dx} + 5y^2\). Substituting \(\frac{dy}{dx}\) and \(y = \cot(5x)\), we get:
\[
\frac{dy}{dx} + 5y^2 = -5\csc^2(5x) + 5(\cot^2(5x))
\]

### Step 3: Simplify the expression

We know from trigonometric identities that:
\[
\csc^2(5x) = 1 + \cot^2(5x)
\]
Thus, the expression becomes:
\[
-5(1 + \cot^2(5x)) + 5\cot^2(5x)
\]
Simplifying further:
\[
-5 - 5\cot^2(5x) + 5\cot^2(5x) = -5
\]

### Final Answer:
\[
\boxed{-5}
\]

Would you like more details or have any questions about this solution? 

Here are five related questions for further practice:
1. How would you differentiate \(y = \sin(5x)\)?
2. What is the derivative of \(y = \sec(3x)\)?
3. How do you simplify \(\sin^2(x) + \cos^2(x)\)?
4. If \(y = \tan(x)\), what is \(\frac{dy}{dx}\)?
5. How do you solve \(\cot^2(x) = 1\)?

**Tip:** Always double-check trigonometric identities while solving differentiation problems!

Solve the problem of finding dy/dx + 5y^2 when y = cot(5x) using differentiation and trigonometric identities. This detailed solution is suitable for high school students in Grades 11-12.

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