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Löse b) ohne sec. F(x) = ln(x) g(x) =(x)^1/2 h(x) = 1+ tan^2(x)

To solve part b) of the problem without using \(\sec(x)\), we proceed step-by-step:

The function given is:
\[
\varphi(x) = \ln\left(\sqrt{1 + \tan^2(x)}\right).
\]

### Step 1: Simplify the function
We know \(\sqrt{1 + \tan^2(x)} = |\sec(x)|\). However, since we are solving without explicitly using \(\sec(x)\), we proceed as follows:
\[
\varphi(x) = \ln\left((1 + \tan^2(x))^{1/2}\right).
\]
Using the logarithmic property \(\ln(a^b) = b \ln(a)\), this becomes:
\[
\varphi(x) = \frac{1}{2} \ln(1 + \tan^2(x)).
\]

### Step 2: Differentiate \(\varphi(x)\)
Now we differentiate \(\varphi(x)\):
\[
\varphi'(x) = \frac{1}{2} \cdot \frac{1}{1 + \tan^2(x)} \cdot \frac{d}{dx} [1 + \tan^2(x)].
\]

### Step 3: Differentiate \(1 + \tan^2(x)\)
The derivative of \(1 + \tan^2(x)\) is:
\[
\frac{d}{dx} [1 + \tan^2(x)] = 2\tan(x)\sec^2(x).
\]

### Step 4: Substitute back
Substituting this into the derivative of \(\varphi(x)\):
\[
\varphi'(x) = \frac{1}{2} \cdot \frac{2\tan(x)}{1 + \tan^2(x)}.
\]

### Step 5: Simplify the result
The \(2\) cancels out, leaving:
\[
\varphi'(x) = \frac{\tan(x)}{1 + \tan^2(x)}.
\]
Using the trigonometric identity \(1 + \tan^2(x) = \sec^2(x)\), this simplifies further to:
\[
\varphi'(x) = \frac{\sin(x)\cos(x)}{\cos^2(x)} = \sin(x),
\]
if simplified directly without referencing \(\sec(x)\) explicitly.

Let me know if you'd like detailed verification or further steps! Here are related questions for practice:

1. Can you compute the derivative of \(\ln(\tan(x))\)?
2. What is the derivative of \(\sqrt{1 + \sin^2(x)}\)?
3. Solve part (c) of the same exercise and compare the approach.
4. Find the derivative of \(\ln(1 + x^2)\).
5. How does the chain rule apply to trigonometric functions generally?

**Tip:** Always simplify logarithmic expressions before differentiating; it saves effort and reduces errors

Learn how to solve the problem involving the derivative of the logarithmic and trigonometric functions. This solution demonstrates how to differentiate a function of the form ln(sqrt(1 + tan^2(x))) without using secant explicitly. The problem is suitable for advanced high school students studying calculus, covering logarithmic differentiation and trigonometric identities.

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