Derivative of an Integral Calculator (FTC Part 2)

Calculate the derivative of a definite integral with variable limits using the Second Fundamental Theorem of Calculus (Leibniz Integral Rule).

Resulting Derivative F'(x):

Calculation Breakdown

Component	Expression
f(b(x))
b'(x)
f(a(x))
a'(x)
F'(x) = f(b(x))b'(x) – f(a(x))a'(x)

What is the Second Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus is a cornerstone theorem that links the concepts of differentiating a function and integrating a function. It is composed of two parts. While the first part is often used to evaluate definite integrals, the second part provides a powerful method for finding the derivative of an integral. Specifically, it tells us how to differentiate a function that is defined as a definite integral.

In its simplest form, the Second Fundamental Theorem of Calculus states that if you have a function G(x) defined by an integral with a variable upper limit, like so:

G(x) = ∫ [from a to x] f(t) dt

Then the derivative of G(x) is simply the original function f(t) with ‘t’ replaced by ‘x’. That is, G'(x) = f(x). This shows that differentiation and integration are inverse processes.

The Formula: Leibniz Integral Rule

The simple case is great, but what happens when both the upper and lower limits of integration are functions of x? To handle this, we use a more general version of the theorem, often called the Leibniz Integral Rule. This is what our calculator uses.

Given a function defined as: F(x) = ∫ [from a(x) to b(x)] f(t) dt

The derivative, F'(x), is found using the formula:

F'(x) = f(b(x)) * b'(x) - f(a(x)) * a'(x)

Formula Variables

Variable	Meaning	Unit	Typical Range
f(t)	The integrand; the function being integrated.	Unitless (mathematical expression)	Any continuous function.
a(x)	The lower limit of integration, as a function of x.	Unitless (mathematical expression)	Any differentiable function.
b(x)	The upper limit of integration, as a function of x.	Unitless (mathematical expression)	Any differentiable function.
a'(x)	The derivative of the lower limit.	Unitless (mathematical expression)	Derivative of a(x).
b'(x)	The derivative of the upper limit.	Unitless (mathematical expression)	Derivative of b(x).

Practical Examples

Let’s walk through two examples to see how the formula works in practice.

Example 1: Simple Upper Limit

Find the derivative of F(x) = ∫ [from 0 to x] cos(t) dt.

• Inputs:
  • f(t) = cos(t)
  • a(x) = 0 => a'(x) = 0
  • b(x) = x => b'(x) = 1
• Calculation:
  • f(b(x)) = cos(x)
  • f(a(x)) = cos(0) = 1
  • F'(x) = cos(x) * (1) - 1 * (0)
• Result: F'(x) = cos(x)

Example 2: Variable Upper and Lower Limits

Find the derivative of F(x) = ∫ [from x to x³] (1/t) dt.

• Inputs:
  • f(t) = 1/t
  • a(x) = x => a'(x) = 1
  • b(x) = x³ => b'(x) = 3x²
• Calculation:
  • f(b(x)) = 1 / (x³)
  • f(a(x)) = 1 / (x)
  • F'(x) = (1/x³) * (3x²) - (1/x) * (1)
• Result: F'(x) = 3/x - 1/x = 2/x

How to Use This Derivative Calculator

Using this tool is straightforward. You simply need to provide the five components of the Leibniz Rule formula.

1. Integrand f(t): Enter the core function you are integrating. Make sure to use ‘t’ as the variable.
2. Upper Limit b(x): Enter the function that defines the upper bound of your integral. Use ‘x’ as the variable.
3. Derivative of Upper Limit b'(x): Manually calculate and enter the derivative of the upper limit function with respect to x.
4. Lower Limit a(x): Enter the function that defines the lower bound of your integral.
5. Derivative of Lower Limit a'(x): Manually calculate and enter the derivative of the lower limit function with respect to x.
6. Calculate: Click the “Calculate Derivative” button. The result F'(x) will be displayed, along with a table showing how each part of the formula was constructed.

Key Factors That Affect the Calculation

• Continuity of f(t): The theorem requires the integrand f(t) to be a continuous function over the interval of integration.
• Differentiability of Limits: Both the upper limit b(x) and the lower limit a(x) must be differentiable functions. If they are not, the rule cannot be applied.
• The Chain Rule: The terms b'(x) and a'(x) are a direct application of the Chain Rule. The theorem is essentially differentiating the antiderivative with respect to the limits, and then differentiating the limits themselves.
• Constant Limits: If a limit is a constant (e.g., a(x) = 5), its derivative is zero (a'(x) = 0). This simplifies the formula significantly, as the corresponding term becomes zero.
• Swapping Limits: If you swap the limits of integration, the result is negated. This is a fundamental property of definite integrals. Flipping the limits from ∫[a, b] to ∫[b, a] results in multiplying the final derivative by -1.
• Complexity of Expressions: The final derivative is constructed symbolically. The complexity of the inputs for f(t), a(x), and b(x) will directly determine the complexity of the output expression.

Frequently Asked Questions (FAQ)

What is the difference between the First and Second Fundamental Theorem of Calculus?

The First FTC relates a definite integral to the net change in its antiderivative, primarily used for evaluating integrals (e.g., `∫[a, b] f(x)dx = F(b) – F(a)`). The Second FTC explains how to find the derivative of an integral function, showing that differentiation and integration are inverse operations.

Why is this also called Leibniz’s Rule?

Gottfried Wilhelm Leibniz, one of the founders of calculus, generalized the theorem to handle cases where the limits of integration are functions themselves. This more general form is widely known as the Leibniz Integral Rule.

What if one of the limits is just ‘x’ and the other is a constant?

This is the simplest non-trivial case. For example, if F(x) = ∫[c, x] f(t) dt, then a(x) = c (a constant) and b(x) = x. Their derivatives are a'(x) = 0 and b'(x) = 1. The formula simplifies to F'(x) = f(x) * 1 - f(c) * 0 = f(x), which is the basic statement of the theorem.

Why do I need to enter the derivatives of the limits myself?

Performing symbolic differentiation automatically requires a sophisticated computer algebra system, which is beyond the scope of a simple browser-based tool. By providing the derivatives manually, you enable the calculator to focus on correctly applying the FTC formula. For more complex derivatives, you can use a standard derivative calculator.

Are there any units involved?

No. This is a calculator for abstract mathematical expressions. The inputs and outputs are functions and are unitless.

What happens if I enter an invalid mathematical expression?

The calculator performs symbolic string replacement. It does not evaluate the math. If you enter a string like “log(t)”, it will replace ‘t’ correctly, but it doesn’t understand what “log” means. The output will be a symbolic representation based on your exact inputs.

Can I use this for definite integrals with numeric answers?

No. This calculator provides a symbolic derivative, which is a new function. It does not evaluate the integral to get a number. For that, you would need an integral calculator.

What is an “integrand”?

The integrand is simply the function that appears inside the integral sign, the one that you are integrating. In the expression ∫ f(t) dt, f(t) is the integrand.

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