y= sin(sinx) d^2y/dx^x+dy/dxtanx+f(x)=0 then f(x) equals to

Given the equation:

\[
y = \sin(\sin x)
\]
\[
\frac{d^2y}{dx^2} + \frac{dy}{dx} \tan x + f(x) = 0
\]

We need to find \( f(x) \). Let's proceed step by step.

### Step 1: Find the first derivative \( \frac{dy}{dx} \)
Since \( y = \sin(\sin x) \), applying the chain rule:

\[
\frac{dy}{dx} = \cos(\sin x) \cdot \frac{d}{dx}(\sin x) = \cos(\sin x) \cdot \cos x
\]

Thus, the first derivative is:

\[
\frac{dy}{dx} = \cos(\sin x) \cdot \cos x
\]

### Step 2: Find the second derivative \( \frac{d^2y}{dx^2} \)
We differentiate \( \frac{dy}{dx} \) again using the product rule:

\[
\frac{d^2y}{dx^2} = \frac{d}{dx}\left( \cos(\sin x) \cdot \cos x \right)
\]

Using the product rule:

\[
\frac{d^2y}{dx^2} = \frac{d}{dx}\left( \cos(\sin x) \right) \cdot \cos x + \cos(\sin x) \cdot \frac{d}{dx}(\cos x)
\]

Now, differentiate each part:
\[
\frac{d}{dx}\left( \cos(\sin x) \right) = -\sin(\sin x) \cdot \cos x
\]
\[
\frac{d}{dx}(\cos x) = -\sin x
\]

Substituting these into the second derivative expression:

\[
\frac{d^2y}{dx^2} = -\sin(\sin x) \cdot \cos^2 x - \cos(\sin x) \cdot \sin x
\]

### Step 3: Substitute into the original equation
Substitute \( \frac{d^2y}{dx^2} \) and \( \frac{dy}{dx} \) into the original equation:

\[
-\sin(\sin x) \cdot \cos^2 x - \cos(\sin x) \cdot \sin x + \cos(\sin x) \cdot \cos x \tan x + f(x) = 0
\]

Simplifying the terms:

\[
-\sin(\sin x) \cdot \cos^2 x - \cos(\sin x) \cdot \sin x + \cos(\sin x) \cdot \sin x + f(x) = 0
\]

Notice that \( \cos(\sin x) \cdot \sin x \) and \( -\cos(\sin x) \cdot \sin x \) cancel out, leaving:

\[
-\sin(\sin x) \cdot \cos^2 x + f(x) = 0
\]

### Step 4: Solve for \( f(x) \)

\[
f(x) = \sin(\sin x) \cdot \cos^2 x
\]

Thus, the function \( f(x) \) is:

\[
f(x) = \sin(\sin x) \cdot \cos^2 x
\]

Would you like further details on any of these steps?

Here are 5 related questions for practice:
1. How would the result change if \( y = \cos(\cos x) \)?
2. What is the third derivative \( \frac{d^3y}{dx^3} \) for \( y = \sin(\sin x) \)?
3. How do you apply the chain rule when differentiating nested trigonometric functions?
4. What happens if the tangent function in the original equation is replaced by \( \cot x \)?
5. How would you find the second derivative of \( f(x) = \sin(\sin x) \cdot \cos^2 x \)?

**Tip:** When differentiating composite functions, always use the chain rule, carefully tracking each derivative step.

y = sin(sin x), d^2y/dx^2 + dy/dx * tan(x) + f(x) = 0, then f(x) equals to

Learn how to find the function f(x) in the equation y = sin(sin x) with a second derivative and trigonometric functions. This step-by-step solution uses the chain rule and product rule. Ideal for advanced high school or college students.

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