give me the derivitive of f(t)= ^4√ t − 1 ^4√ t

Let's find the derivative of the function \( f(t) = \sqrt[4]{t} - \frac{1}{\sqrt[4]{t}} \).

We can rewrite \( \sqrt[4]{t} \) and \( \frac{1}{\sqrt[4]{t}} \) using exponents:

\[
f(t) = t^{1/4} - t^{-1/4}
\]

### Step 1: Differentiate each term
- The derivative of \( t^{1/4} \) is found using the power rule \( \frac{d}{dt} t^n = n t^{n-1} \):
  \[
  \frac{d}{dt} t^{1/4} = \frac{1}{4} t^{-3/4}
  \]

- The derivative of \( t^{-1/4} \) is:
  \[
  \frac{d}{dt} t^{-1/4} = -\frac{1}{4} t^{-5/4}
  \]

### Step 2: Combine the results
Now, we combine both derivatives:
\[
f'(t) = \frac{1}{4} t^{-3/4} + \frac{1}{4} t^{-5/4}
\]

### Final answer:
\[
f'(t) = \frac{1}{4} t^{-3/4} + \frac{1}{4} t^{-5/4}
\]

Do you want more details or have any questions?

Here are some related questions:
1. What are the power rules for derivatives?
2. How do we differentiate functions with fractional exponents?
3. What is the significance of negative exponents in derivatives?
4. Can this derivative be simplified further?
5. What other rules of differentiation apply to fractional powers?

**Tip:** When dealing with radicals, converting them to exponents can make differentiation easier.

give me the derivative of f(t)= ^4√ t − 1 ^4√ t

Learn how to differentiate the function f(t) = √[4]{t} − 1/√[4]{t} using the power rule and handling exponents. This detailed solution covers the key calculus concepts of exponents and the power rule, suitable for students in Grades 11-12 or early college.

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