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The given problem in the image asks to find \( f\left(\frac{1}{78}\right) + f'\left(\frac{1}{78}\right) \), where \( f(x) \) is defined as:

\[
f(x) = \int_1^{78x} \frac{dt}{(t^2 + 1)^3}
\]

### Step 1: Find \( f\left(\frac{1}{78}\right) \)

To find \( f\left(\frac{1}{78}\right) \), we substitute \( x = \frac{1}{78} \) into the expression for \( f(x) \). Thus, we need to compute:

\[
f\left(\frac{1}{78}\right) = \int_1^1 \frac{dt}{(t^2 + 1)^3}
\]

The limits of integration are now the same (both 1), so the integral evaluates to 0:

\[
f\left(\frac{1}{78}\right) = 0
\]

### Step 2: Find \( f'(x) \)

To differentiate \( f(x) \), we apply the Fundamental Theorem of Calculus and the Chain Rule. We know that the derivative of an integral of the form:

\[
f(x) = \int_a^{g(x)} h(t)\,dt
\]

is:

\[
f'(x) = h(g(x)) \cdot g'(x)
\]

In this case, \( g(x) = 78x \) and \( h(t) = \frac{1}{(t^2 + 1)^3} \). Therefore, we get:

\[
f'(x) = \frac{1}{((78x)^2 + 1)^3} \cdot 78
\]

Simplifying:

\[
f'(x) = \frac{78}{(6084x^2 + 1)^3}
\]

### Step 3: Find \( f'\left(\frac{1}{78}\right) \)

Now, substitute \( x = \frac{1}{78} \) into \( f'(x) \):

\[
f'\left(\frac{1}{78}\right) = \frac{78}{(6084\left(\frac{1}{78}\right)^2 + 1)^3}
\]

Simplify the expression inside the parentheses:

\[
f'\left(\frac{1}{78}\right) = \frac{78}{(6084 \cdot \frac{1}{6084} + 1)^3} = \frac{78}{(1 + 1)^3} = \frac{78}{8}
\]

Thus:

\[
f'\left(\frac{1}{78}\right) = 9.75
\]

### Step 4: Final Result

The total sum is:

\[
f\left(\frac{1}{78}\right) + f'\left(\frac{1}{78}\right) = 0 + 9.75 = 9.75
\]

### Final Answer:

\[
\boxed{9.75}
\]

Let me know if you'd like further details or have any questions!

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Here are 5 related questions to deepen your understanding:
1. How does the Fundamental Theorem of Calculus apply to finding derivatives of integrals?
2. What is the Chain Rule, and how is it applied in this problem?
3. How would the result change if the limits of integration were not identical after substitution?
4. What happens to the integral when the upper and lower limits are swapped?
5. Can this method be extended to find \( f(x) \) and \( f'(x) \) for more complicated functions?

**Tip**: When evaluating definite integrals where the limits are the same, the result will always be zero regardless of the integrand!

Given that f(x) = ∫ from 1 to 78x (dt / (t^2 + 1)^3), find f(1/78) + f'(1/78).

Learn how to solve the definite integral and find f(1/78) + f'(1/78), using the Fundamental Theorem of Calculus and Chain Rule. This advanced calculus problem is ideal for Grades 11-12 students.

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