Derivative of the Integral Calculator | Leibniz Rule

Derivative of the Integral Calculator

This calculator finds the derivative of an integral with variable limits using the Leibniz Integral Rule. Enter the function f(t), the limits a(x) and b(x), and their derivatives a'(x) and b'(x) to find the derivative of the integral from a(x) to b(x) of f(t) dt with respect to x.

Calculator

Function f(t): Enter the function to be integrated, using 't' as the variable (e.g., t*t, Math.sin(t), Math.exp(t)).

Lower Limit a(x): Enter the lower limit of integration as a function of 'x' (e.g., x, 2*x, 5).

Derivative a'(x): Enter the derivative of a(x) with respect to 'x' (e.g., 1, 2, 0).

Upper Limit b(x): Enter the upper limit of integration as a function of 'x' (e.g., x*x, Math.sin(x), 10).

Derivative b'(x): Enter the derivative of b(x) with respect to 'x' (e.g., 2*x, Math.cos(x), 0).

Value of x for Evaluation: Enter the value of 'x' at which to evaluate the derivative.

Chart X-Range (+/-): Range around x_val for the chart (x_val – range to x_val + range).

Chart Points: Number of points to plot on the chart (odd number, 3-101).

Result will appear here.

f(b(x)) expression: –

f(a(x)) expression: –

Derivative at x=2: –

f(b(x_val)) * b'(x_val) at x=2: –

f(a(x_val)) * a'(x_val) at x=2: –

Formula: d/dx [∫(from a(x) to b(x)) f(t) dt] = f(b(x)) * b'(x) – f(a(x)) * a'(x)

Derivative Chart

Chart showing f(b(x))b'(x), f(a(x))a'(x), and the derivative value around x=2.

What is the Derivative of the Integral Calculator?

The derivative of the integral calculator is a tool used to find the derivative of a definite integral where the limits of integration are functions of the variable with respect to which we are differentiating. This process is governed by the Leibniz Integral Rule, which is an extension of the Fundamental Theorem of Calculus. Our derivative of the integral calculator applies this rule.

Specifically, if you have an integral of the form:
I(x) = ∫_a(x)^b(x) f(t) dt
The calculator finds dI/dx. This is useful in various fields like physics, engineering, and economics where the bounds of integration might change. The derivative of the integral calculator simplifies this complex differentiation.

Who should use it? Students of calculus, engineers, physicists, and anyone dealing with integrals with variable limits will find this derivative of the integral calculator helpful. Common misconceptions include thinking it's the same as just evaluating f(x) (which is only true if the limits are 0 to x and f is continuous).

Derivative of the Integral Formula and Mathematical Explanation

The derivative of an integral with variable limits is found using the Leibniz Integral Rule. Given an integral:

I(x) = ∫_a(x)^b(x) f(t) dt

Its derivative with respect to x is given by:

d/dx [I(x)] = d/dx [∫_a(x)^b(x) f(t) dt] = f(b(x)) * b'(x) – f(a(x)) * a'(x)

Where:

f(t) is the integrand.
a(x) is the lower limit of integration.
b(x) is the upper limit of integration.
a'(x) is the derivative of the lower limit with respect to x.
b'(x) is the derivative of the upper limit with respect to x.

The derivative of the integral calculator implements this formula directly.

Variables Table

Variable	Meaning	Unit	Typical Input
f(t)	The function being integrated (integrand)	Varies	Expression like t*t, Math.sin(t)
a(x)	Lower limit of integration	Varies	Expression like x, 2*x, 5
b(x)	Upper limit of integration	Varies	Expression like x*x, Math.sin(x)
a'(x)	Derivative of a(x) w.r.t. x	Varies	Expression like 1, 2, Math.cos(x)
b'(x)	Derivative of b(x) w.r.t. x	Varies	Expression like 2*x, Math.cos(x)
x	Variable for limits and differentiation	Varies	Number for evaluation
t	Variable of integration	Varies	Used in f(t)

Variables used in the Leibniz Integral Rule.

Practical Examples

Let's see how our derivative of the integral calculator works with examples.

Example 1:

Find d/dx [∫_x^x² t² dt]

Here, f(t) = t², a(x) = x, b(x) = x². So, a'(x) = 1, b'(x) = 2x. f(b(x)) = f(x²) = (x²)² = x⁴ f(a(x)) = f(x) = x² Using the formula: f(b(x))b'(x) – f(a(x))a'(x) = x⁴ * (2x) – x² * (1) = 2x⁵ – x². If x=2, the value is 2*(2^5) – 2^2 = 2*32 – 4 = 64 – 4 = 60. Our derivative of the integral calculator can verify this.

Example 2:

Find d/dx [∫₀^sin(x) e^t dt]

Here, f(t) = e^t, a(x) = 0, b(x) = sin(x). So, a'(x) = 0, b'(x) = cos(x). f(b(x)) = f(sin(x)) = e^sin(x) f(a(x)) = f(0) = e⁰ = 1 Using the formula: f(b(x))b'(x) – f(a(x))a'(x) = e^sin(x) * cos(x) – 1 * 0 = e^sin(x)cos(x). The derivative of the integral calculator can handle such functions if entered as Math.exp(t), Math.sin(x), Math.cos(x).

How to Use This Derivative of the Integral Calculator

Enter f(t): Input the integrand function using 't' as the variable (e.g., `t*t`, `Math.exp(t)`).
Enter a(x) and a'(x): Input the lower limit a(x) as a function of 'x' and its derivative a'(x).
Enter b(x) and b'(x): Input the upper limit b(x) as a function of 'x' and its derivative b'(x).
Enter x value: Provide the 'x' value at which you want to evaluate the derivative.
Enter Chart Range and Points: Specify the range around x and the number of points for the chart.
View Results: The calculator automatically displays the derivative expression, its value at x, and intermediate terms. The chart is also updated.

The derivative of the integral calculator provides both the symbolic form (before simplification) and a numerical value.

Key Factors That Affect Derivative of the Integral Results

The form of f(t): The complexity of the integrand directly influences f(a(x)) and f(b(x)).
The lower limit a(x): How the lower limit changes with x affects the f(a(x))a'(x) term.
The upper limit b(x): How the upper limit changes with x affects the f(b(x))b'(x) term.
The derivatives a'(x) and b'(x): These represent the rates of change of the limits and scale the f(a(x)) and f(b(x)) terms.
The point of evaluation x: The specific value of x determines the numerical result of the derivative.
Continuity and Differentiability: The Leibniz rule requires f(t) to be continuous and a(x), b(x) to be differentiable.

Frequently Asked Questions (FAQ)

What is the Leibniz Integral Rule?: It's a rule for differentiating under the integral sign, especially when the limits of integration are functions of the variable of differentiation. Our derivative of the integral calculator is based on this.
What if the limits are constants?: If a(x)=a and b(x)=b (constants), then a'(x)=0 and b'(x)=0, so the derivative of the integral is 0, as expected for a constant value.
Is this related to the Fundamental Theorem of Calculus?: Yes, the Leibniz rule is a generalization of the Fundamental Theorem of Calculus, Part 1, which deals with d/dx ∫_a^x f(t) dt = f(x).
Can the calculator handle any function f(t), a(x), b(x)?: The calculator can handle functions expressible in standard JavaScript mathematical notation (using `Math.` prefix for functions like `sin`, `cos`, `exp`, `pow`, `log`). You must also provide the derivatives a'(x) and b'(x) manually.
Why do I need to enter a'(x) and b'(x)?: This calculator does not perform symbolic differentiation. You need to calculate the derivatives of your limit functions a(x) and b(x) and enter them.
What if f(t) also depends on x?: The version of the Leibniz rule used here assumes f(t) is only a function of t. If f(t,x) is integrated, a more general rule is needed involving partial derivatives, which this calculator does not handle.
How accurate is the numerical evaluation?: It uses standard JavaScript floating-point arithmetic, which is generally accurate for most practical purposes but subject to precision limitations.
Where can I learn more about the Leibniz integral rule?: You can find more information in calculus textbooks or online resources discussing differentiation under the integral sign and the Leibniz integral rule.

Find The Derivative Of The Integral Calculator