Key Concepts and Formulas

• Fundamental Theorem of Calculus (Part I): If $F(x) = \int_a^x g(t) dt$, where $a$ is a constant, then $F'(x) = \frac{d}{dx} \left( \int_a^x g(t) dt \right) = g(x)$.
• Leibniz Integral Rule (for differentiation of integrals with variable limits): If $G(x) = \int_{a(x)}^{b(x)} h(t) dt$, then $G'(x) = h(b(x)) \cdot b'(x) - h(a(x)) \cdot a'(x)$.
• Quotient Rule for Differentiation: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$.

Step-by-Step Solution

Step 1: Understand the Given Equation and the Goal We are given the equation: $\int\limits_0^x f(t) dt = x^2 + \int\limits_x^1 t^2 f(t) dt$ Our objective is to find the value of $f'\left( \frac{1}{2} \right)$. To achieve this, we will first find an explicit expression for $f(x)$ by differentiating the given equation with respect to $x$, and then differentiate $f(x)$ to find $f'(x)$.

Step 2: Differentiate Both Sides of the Equation with Respect to $x$ We apply the differentiation operator $\frac{d}{dx}$ to both sides of the given equation.

• Left Hand Side (LHS): The LHS is $\int\limits_0^x f(t) dt$. By the Fundamental Theorem of Calculus (Part I), its derivative with respect to $x$ is $f(x)$. $\frac{d}{dx} \left( \int\limits_0^x f(t) dt \right) = f(x)$

• Right Hand Side (RHS): The RHS is $x^2 + \int\limits_x^1 t^2 f(t) dt$. We differentiate each term separately.

  • The derivative of $x^2$ is $\frac{d}{dx}(x^2) = 2x$.
  • For the integral term $\int\limits_x^1 t^2 f(t) dt$, we use the Leibniz Integral Rule. Here, the upper limit is $b(x) = 1$ (a constant), and the lower limit is $a(x) = x$. The integrand is $h(t) = t^2 f(t)$. The derivatives of the limits are $b'(x) = \frac{d}{dx}(1) = 0$ and $a'(x) = \frac{d}{dx}(x) = 1$. Applying the rule: $\frac{d}{dx} \left( \int\limits_x^1 t^2 f(t) dt \right) = h(b(x)) \cdot b'(x) - h(a(x)) \cdot a'(x)$ $= (1^2 f(1)) \cdot 0 - (x^2 f(x)) \cdot 1$ $= 0 - x^2 f(x) = -x^2 f(x)$

• Equating the Derivatives: Now, we equate the derivatives of the LHS and RHS: $f(x) = 2x - x^2 f(x)$

Step 3: Solve for $f(x)$ We rearrange the equation obtained in Step 2 to isolate $f(x)$: $f(x) + x^2 f(x) = 2x$ Factor out $f(x)$ from the terms on the left side: $f(x) (1 + x^2) = 2x$ Divide by $(1 + x^2)$ to get the expression for $f(x)$: $f(x) = \frac{2x}{1 + x^2}$

Step 4: Find the Derivative of $f(x)$, i.e., $f'(x)$ We have $f(x) = \frac{2x}{1 + x^2}$. To find $f'(x)$, we use the Quotient Rule. Let $u(x) = 2x$ and $v(x) = 1 + x^2$. Then, $u'(x) = \frac{d}{dx}(2x) = 2$ and $v'(x) = \frac{d}{dx}(1 + x^2) = 2x$. Applying the Quotient Rule $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}$: $f'(x) = \frac{(2)(1 + x^2) - (2x)(2x)}{(1 + x^2)^2}$ $f'(x) = \frac{2 + 2x^2 - 4x^2}{(1 + x^2)^2}$ $f'(x) = \frac{2 - 2x^2}{(1 + x^2)^2}$

Step 5: Evaluate $f'\left( \frac{1}{2} \right)$ Substitute $x = \frac{1}{2}$ into the expression for $f'(x)$: $f'\left( \frac{1}{2} \right) = \frac{2 - 2\left(\frac{1}{2}\right)^2}{\left(1 + \left(\frac{1}{2}\right)^2\right)^2}$ $f'\left( \frac{1}{2} \right) = \frac{2 - 2\left(\frac{1}{4}\right)}{\left(1 + \frac{1}{4}\right)^2}$ $f'\left( \frac{1}{2} \right) = \frac{2 - \frac{1}{2}}{\left(\frac{5}{4}\right)^2}$ $f'\left( \frac{1}{2} \right) = \frac{\frac{3}{2}}{\frac{25}{16}}$ To simplify the complex fraction, we multiply the numerator by the reciprocal of the denominator: $f'\left( \frac{1}{2} \right) = \frac{3}{2} \times \frac{16}{25}$ $f'\left( \frac{1}{2} \right) = \frac{3 \times 8}{25}$ $f'\left( \frac{1}{2} \right) = \frac{24}{25}$

This value matches option (C).

Common Mistakes & Tips

• Sign Error with Lower Limit: When differentiating an integral with $x$ as the lower limit (e.g., $\int_x^a g(t) dt$), remember the derivative is $-g(x)$. This is a common source of error.
• Algebraic Precision: Ensure accuracy in algebraic manipulations, especially when combining terms and simplifying fractions.
• Leibniz Rule Application: Correctly identify the upper and lower limit functions and their derivatives when applying the Leibniz Integral Rule.

Summary

The problem requires applying the Fundamental Theorem of Calculus and the Leibniz Integral Rule to differentiate the given integral equation. This process yields an explicit expression for $f(x)$. Subsequently, the Quotient Rule is used to find the derivative $f'(x)$, which is then evaluated at $x = \frac{1}{2}$ to obtain the final answer.

The final answer is $\boxed{\frac{24}{25}}$.