If  
f
(
x
)
=
∫
x
2
x
t
4
d
t
 
then
f
'
(
x
)
= what?

The given function is:

\[
f(x) = \int_{x^2}^{x} t^4 \, dt
\]

To find \( f'(x) \), we need to apply the Leibniz rule for differentiating an integral with variable limits of integration. This rule is given as:

\[
\frac{d}{dx} \left( \int_{a(x)}^{b(x)} g(t) \, dt \right) = g(b(x)) \cdot b'(x) - g(a(x)) \cdot a'(x)
\]

In this case:

- \( g(t) = t^4 \)
- \( a(x) = x^2 \), so \( a'(x) = 2x \)
- \( b(x) = x \), so \( b'(x) = 1 \)

Using the Leibniz rule:

\[
f'(x) = t^4 \bigg|_{t=x} \cdot (1) - t^4 \bigg|_{t=x^2} \cdot (2x)
\]

This simplifies to:

\[
f'(x) = x^4 - (x^2)^4 \cdot 2x
\]

\[
f'(x) = x^4 - 2x \cdot x^8
\]

\[
f'(x) = x^4 - 2x^9
\]

Thus, the derivative is:

\[
f'(x) = x^4 - 2x^9
\]

Do you want more details or have any questions?

Here are 5 related questions for further practice:
1. What is the derivative of \( \int_0^x t^4 \, dt \)?
2. Can you apply the Leibniz rule for an integral where both limits depend on \(x\)?
3. What happens if \( f(x) = \int_1^{x^3} \sin(t) \, dt \)? What is \( f'(x) \)?
4. How would you compute \( f'(x) \) if the integrand was \( t^n \)?
5. What is the integral of \( t^4 \) with respect to \(t\)?

**Tip**: When differentiating an integral with variable limits, always apply the Leibniz rule carefully to account for how the limits depend on \(x\).

If f(x) = ∫(x² to x) t⁴ dt, then f'(x) = what?

Explore how to differentiate the integral f(x) = ∫(x² to x) t⁴ dt using the Leibniz rule. This solution explains step-by-step how to apply the rule for variable limits, including a detailed breakdown of each term in the derivative.

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f ( x )

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