Here are five examples of derivatives using the **chain rule**:

### **Example 1**  
Find the derivative of:  
\[
f(x) = (3x^2 + 5)^4
\]  
#### **Solution**  
Let \( u = 3x^2 + 5 \), then \( f(x) = u^4 \).  
Using the chain rule:  
\[
\frac{d}{dx} u^4 = 4u^3 \cdot \frac{du}{dx}
\]  
Since \( \frac{du}{dx} = 6x \), we get:  
\[
f'(x) = 4(3x^2 + 5)^3 \cdot 6x = 24x(3x^2 + 5)^3
\]

---

### **Example 2**  
Find the derivative of:  
\[
f(x) = \sin(5x^3 + 2x)
\]  
#### **Solution**  
Let \( u = 5x^3 + 2x \), then \( f(x) = \sin u \).  
Using the chain rule:  
\[
\frac{d}{dx} \sin u = \cos u \cdot \frac{du}{dx}
\]  
Since \( \frac{du}{dx} = 15x^2 + 2 \), we get:  
\[
f'(x) = \cos(5x^3 + 2x) \cdot (15x^2 + 2)
\]

---

### **Example 3**  
Find the derivative of:  
\[
f(x) = e^{\sqrt{x^2 + 1}}
\]  
#### **Solution**  
Let \( u = \sqrt{x^2 + 1} \), then \( f(x) = e^u \).  
Using the chain rule:  
\[
\frac{d}{dx} e^u = e^u \cdot \frac{du}{dx}
\]  
Since \( u = (x^2 + 1)^{1/2} \), its derivative is:  
\[
\frac{du}{dx} = \frac{1}{2}(x^2 + 1)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + 1}}
\]  
Thus,  
\[
f'(x) = e^{\sqrt{x^2 + 1}} \cdot \frac{x}{\sqrt{x^2 + 1}}
\]

---

### **Example 4**  
Find the derivative of:  
\[
f(x) = \ln(2x^2 + 3x + 1)
\]  
#### **Solution**  
Let \( u = 2x^2 + 3x + 1 \), then \( f(x) = \ln u \).  
Using the chain rule:  
\[
\frac{d}{dx} \ln u = \frac{1}{u} \cdot \frac{du}{dx}
\]  
Since \( \frac{du}{dx} = 4x + 3 \), we get:  
\[
f'(x) = \frac{4x + 3}{2x^2 + 3x + 1}
\]

---

### **Example 5**  
Find the derivative of:  
\[
f(x) = (x^3 + 4x + 7)^{10}
\]  
#### **Solution**  
Let \( u = x^3 + 4x + 7 \), then \( f(x) = u^{10} \).  
Using the chain rule:  
\[
\frac{d}{dx} u^{10} = 10u^9 \cdot \frac{du}{dx}
\]  
Since \( \frac{du}{dx} = 3x^2 + 4 \), we get:  
\[
f'(x) = 10(x^3 + 4x + 7)^9 \cdot (3x^2 + 4)
\]

---

Would you like further explanation on any of these?  

### **Related Questions:**  
1. How does the chain rule relate to implicit differentiation?  
2. Can the chain rule be applied multiple times in a function?  
3. What is the difference between the product rule and the chain rule?  
4. How do you apply the chain rule when differentiating trigonometric functions?  
5. How does the chain rule extend to partial derivatives in multivariable calculus?  

**Tip:** Always identify the inner function before applying the chain rule!

Explore 5 examples of differentiating functions using the chain rule in calculus. Each example includes a step-by-step explanation to help you understand how to apply the chain rule to complex functions involving polynomials, trigonometric functions, exponentials, and logarithms. Ideal for high school students and calculus learners.

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