# Chain Rule: Derivatives of the Form $e^{g(x)}$

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### **Example 1:**
Find the derivative of
$$f(x) = e^{2x}$$

**Solution:**
- Here, $g(x) = 2x$, so $g'(x) = 2$.
- Using the chain rule:
$$ f'(x) = e^{2x} \cdot 2 = 2e^{2x} $$

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### **Example 2:**
Find the derivative of
$$ f(x) = e^{x^2} $$

**Solution:**
- Here, $g(x) = x^2$, so $g'(x) = 2x$.
- Using the chain rule:
$$ f'(x) = e^{x^2} \cdot 2x = 2x e^{x^2} $$

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### **Example 3:**
Find the derivative of
$$ f(x) = e^{3x^3 + x} $$

**Solution:**
- Here, $g(x) = 3x^3 + x$, so $g'(x) = 9x^2 + 1$.
- Using the chain rule:
$$ f'(x) = e^{3x^3 + x} \cdot (9x^2 + 1) = (9x^2 + 1) e^{3x^3 + x} $$

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### **Example 4:**
Find the derivative of
$$ f(x) = e^{\frac{1}{x}} $$

**Solution:**
- Here, $g(x) = \frac{1}{x} = x^{-1}$, so $g'(x) = -x^{-2}$.
- Using the chain rule:
$$ f'(x) = e^{x^{-1}} \cdot (-x^{-2}) = -\frac{1}{x^2} e^{\frac{1}{x}} $$

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### **Example 5:**
Find the derivative of
$$ f(x) = e^{\sqrt{x}} $$

**Solution:**
- Here, $g(x) = \sqrt{x} = x^{1/2}$, so $g'(x) = \frac{1}{2}x^{-1/2}$.
- Using the chain rule:
$$ f'(x) = e^{\sqrt{x}} \cdot \frac{1}{2}x^{-1/2} = \frac{e^{\sqrt{x}}}{2\sqrt{x}} $$

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### **Example 6:**
Find the derivative of
$$ f(x) = e^{\ln(x)} $$

**Solution:**
- Here, $g(x) = \ln(x)$, so $g'(x) = \frac{1}{x}$.
- Using the chain rule:
$$ f'(x) = e^{\ln(x)} \cdot \frac{1}{x} = \frac{e^{\ln(x)}}{x}=\dfrac{x}{x}=1 $$

(Recall that $e^{\ln(x)} = x$, so this derivative simplifies to $f'(x) = 1$.)

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### **Example 7:**
Find the derivative of
$$ f(x) = e^{-2x^3} $$

**Solution:**
- Here, $g(x) = -2x^3$, so $g'(x) = -6x^2$.
- Using the chain rule:
$$ f'(x) = e^{-2x^3} \cdot (-6x^2) = -6x^2 e^{-2x^3} $$

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### **Example 8:**
Find the derivative of
$$ f(x) = e^{\frac{x^2}{x+1}} $$

**Solution:**
- Here, $g(x) = \frac{x^2}{x+1}$.
- First, find $g'(x)$ using the quotient rule:
$$ g'(x) = \frac{(2x)(x+1) - (x^2)(1)}{(x+1)^2} = \frac{2x^2 + 2x - x^2}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2} $$
- Using the chain rule:
$$ f'(x) = e^{\frac{x^2}{x+1}} \cdot \frac{x^2 + 2x}{(x+1)^2} $$

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### **Example 9:**
Find the derivative of
$$ f(x) = e^{x^5 + 3x^2} $$

**Solution:**
- Here, $g(x) = x^5 + 3x^2$, so $g'(x) = 5x^4 + 6x$.
- Using the chain rule:
$$ f'(x) = e^{x^5 + 3x^2} \cdot (5x^4 + 6x) = (5x^4 + 6x) e^{x^5 + 3x^2} $$

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### **Example 10:**
Find the derivative of
$$ f(x) = e^{x^{-2}} $$

**Solution:**
- Here, $g(x) = x^{-2}$, so $g'(x) = -2x^{-3}$.
- Using the chain rule:
$$ f'(x) = e^{x^{-2}} \cdot (-2x^{-3}) = -2x^{-3} e^{x^{-2}} $$

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# Summary
For derivatives of the form $e^{g(x)}$, use the chain rule:
$$ \frac{d}{dx}[e^{g(x)}] = e^{g(x)} \cdot g'(x) $$

You first differentiate $g(x)$, then multiply by $e^{g(x)}$. Make sure to correctly apply the chain rule to handle more complex $g(x)$ functions.