Differentiate each function with respect to the given variable. f(w)=2^4 root of w

Let's differentiate the given function \( f(w) = 2^{\sqrt[4]{w}} \) with respect to \( w \).

### Step-by-step solution:

1. **Rewrite the function:**
   \[
   f(w) = 2^{w^{1/4}}
   \]
   Here, \( w^{1/4} \) is the fourth root of \( w \).

2. **Use the chain rule for derivatives:**
   To differentiate \( 2^{w^{1/4}} \), we first apply the general rule for differentiating an exponential function of the form \( a^{g(w)} \), where \( a \) is a constant and \( g(w) \) is a function of \( w \):
   \[
   \frac{d}{dw} \left( a^{g(w)} \right) = a^{g(w)} \ln(a) \cdot \frac{d}{dw} \left( g(w) \right)
   \]
   Here, \( a = 2 \) and \( g(w) = w^{1/4} \).

3. **Differentiate \( g(w) = w^{1/4} \):**
   The derivative of \( w^{1/4} \) with respect to \( w \) is:
   \[
   \frac{d}{dw} \left( w^{1/4} \right) = \frac{1}{4} w^{-3/4}
   \]

4. **Apply the chain rule:**
   Now, combine everything:
   \[
   \frac{d}{dw} f(w) = 2^{w^{1/4}} \ln(2) \cdot \frac{1}{4} w^{-3/4}
   \]

### Final answer:
\[
\frac{d}{dw} f(w) = \frac{\ln(2)}{4} w^{-3/4} 2^{w^{1/4}}
\]

Would you like any further explanation or details?

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Here are 5 related questions to consider:

1. What is the derivative of \( f(w) = 3^{\sqrt[3]{w}} \)?
2. How do we apply the chain rule to differentiate composite functions like \( a^{g(w)} \)?
3. What happens when the base of the exponential is \( e \) instead of \( 2 \)?
4. Can you find the second derivative of \( f(w) = 2^{w^{1/4}} \)?
5. How does the power rule apply to fractional exponents like \( w^{1/4} \)?

**Tip:** Always check if a function can be rewritten in a more convenient form before applying differentiation rules.

Learn how to differentiate the function f(w) = 2^(w^(1/4)) using the chain rule and power rule for fractional exponents. This detailed solution walks through key steps, including applying the chain rule to exponential functions. Suitable for high school students in Grades 11-12.

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