https://file.aiquickdraw.com/mathbot/file/1726788309465b5cpm227.jpg

Let's solve the derivatives for each function in the image.

### 2. Differentiate the following functions with respect to \( x \):

#### (a) \( \sqrt{1 - x^2} \)
We can rewrite the function as:
\[
f(x) = (1 - x^2)^{1/2}
\]
Using the chain rule:
\[
\frac{d}{dx}[(1 - x^2)^{1/2}] = \frac{1}{2}(1 - x^2)^{-1/2} \cdot (-2x) = \frac{-x}{\sqrt{1 - x^2}}
\]

#### (b) \( \sin(3x^3) \)
Using the chain rule:
\[
f(x) = \sin(3x^3)
\]
The derivative of \( \sin(u) \) is \( \cos(u) \), and the derivative of \( 3x^3 \) is \( 9x^2 \). Thus:
\[
\frac{d}{dx}[\sin(3x^3)] = \cos(3x^3) \cdot 9x^2
\]

#### (c) \( \ln(\cos(x)) \)
Using the chain rule:
\[
f(x) = \ln(\cos(x))
\]
The derivative of \( \ln(u) \) is \( \frac{1}{u} \), and the derivative of \( \cos(x) \) is \( -\sin(x) \). Thus:
\[
\frac{d}{dx}[\ln(\cos(x))] = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x)
\]

#### (d) \( \exp(\ln(x)) \)
We simplify first:
\[
f(x) = \exp(\ln(x)) = x
\]
The derivative of \( x \) is simply:
\[
\frac{d}{dx}[x] = 1
\]

#### (e) \( \cos^2(x) \)
We can rewrite \( \cos^2(x) \) as \( (\cos(x))^2 \). Using the chain rule and the power rule:
\[
f(x) = (\cos(x))^2
\]
The derivative of \( u^2 \) is \( 2u \cdot \frac{du}{dx} \), and the derivative of \( \cos(x) \) is \( -\sin(x) \). Thus:
\[
\frac{d}{dx}[\cos^2(x)] = 2\cos(x) \cdot (-\sin(x)) = -2\cos(x)\sin(x) = -\sin(2x)
\]
(using the double-angle identity \( \sin(2x) = 2\sin(x)\cos(x) \)).

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If you would like more detailed explanations or any additional questions answered, feel free to ask!

Here are 5 related questions to expand on this topic:
1. What is the derivative of \( \cos(3x^2) \)?
2. How do you differentiate \( \ln(1 + x^2) \)?
3. Find the derivative of \( \tan^{-1}(x^2) \).
4. What is the derivative of \( e^{x^2} \)?
5. Can you differentiate \( \sin^3(x) \)?

**Tip**: Always simplify functions if possible before differentiating to avoid unnecessary complexity.

Differentiate the following functions with respect to x: (a) sqrt(1 - x^2) (b) sin(3x^3) (c) ln(cos(x)) (d) exp(ln(x)) (e) cos^2(x)

Learn how to differentiate complex functions such as sqrt(1 - x^2), sin(3x^3), ln(cos(x)), exp(ln(x)), and cos^2(x) using key calculus techniques like the chain rule and logarithmic differentiation. Suitable for undergraduate calculus students.

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