Find The Derivative Of An Integral Calculator

Calculate d/dx ∫ f(t) dt

This calculator finds the derivative of an integral with variable limits using the Fundamental Theorem of Calculus and Leibniz rule: d/dx ∫_a(x)^b(x) f(t) dt = f(b(x))b'(x) – f(a(x))a'(x).

Intermediate Values at x

Variable	Expression	Value at x
b(x)
a(x)
f(b(x))
f(a(x))
b'(x)
a'(x)

What is a Derivative of an Integral Calculator?

A derivative of an 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 primarily governed by the Fundamental Theorem of Calculus Part 1 and the Leibniz integral rule for differentiating under the integral sign. Our derivative of an integral calculator applies this rule: d/dx ∫_a(x)^b(x) f(t) dt = f(b(x))b'(x) – f(a(x))a'(x).

This type of calculator is useful for students of calculus, engineers, physicists, and anyone dealing with functions defined by integrals with variable limits. It helps in quickly evaluating such derivatives without manual computation, especially when the functions f(t), a(x), and b(x) are complex.

Common misconceptions involve confusing this with simply integrating f(t) and then differentiating, which only works if the limits are constant or one is variable and the other constant and f(t) is simple. The derivative of an integral calculator handles the more general case where both limits can be functions of x.

Derivative of an Integral Formula and Mathematical Explanation

The derivative of an integral with variable limits is found using an extension of the Fundamental Theorem of Calculus, often combined with the chain rule, which is generalized by the Leibniz integral rule. If we have an integral F(x) defined as:

F(x) = ∫_a(x)^b(x) f(t) dt

Then, its derivative with respect to x, F'(x) or dF/dx, is given by:

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, which is a function of x.
• b(x) is the upper limit of integration, which is a function of x.
• a'(x) is the derivative of a(x) with respect to x.
• b'(x) is the derivative of b(x) with respect to x.
• f(b(x)) is the integrand f(t) evaluated at t = b(x).
• f(a(x)) is the integrand f(t) evaluated at t = a(x).

This formula essentially applies the chain rule along with the Fundamental Theorem of Calculus. The theorem states that if G(t) is an antiderivative of f(t), then ∫_a^b f(t) dt = G(b) – G(a). If a and b are functions of x, we have G(b(x)) – G(a(x)), and differentiating with respect to x gives g(b(x))b'(x) – g(a(x))a'(x), which is f(b(x))b'(x) – f(a(x))a'(x).

Variables Table

Variables used in the derivative of an integral calculation.

Variable	Meaning	Unit	Typical Range
f(t)	Integrand function	Depends on context	Mathematical expressions
a(x)	Lower limit function	Depends on x	Mathematical expressions
b(x)	Upper limit function	Depends on x	Mathematical expressions
a'(x)	Derivative of lower limit	Depends on x	Mathematical expressions
b'(x)	Derivative of upper limit	Depends on x	Mathematical expressions
x	Variable for differentiation	Depends on context	Real numbers
F'(x)	Derivative of the integral	Depends on f and x	Real numbers

Practical Examples (Real-World Use Cases)

Let's see how our derivative of an integral calculator can be used with some examples.

Example 1: Polynomial Functions

Suppose we want to find the derivative of F(x) = ∫_x^x² (t² + 1) dt at x = 2.

• f(t) = t² + 1
• b(x) = x² => b'(x) = 2x
• a(x) = x => a'(x) = 1
• x = 2

At x=2:

• b(2) = 2² = 4
• a(2) = 2
• f(b(2)) = f(4) = 4² + 1 = 17
• f(a(2)) = f(2) = 2² + 1 = 5
• b'(2) = 2 * 2 = 4
• a'(2) = 1

Derivative = f(b(2))b'(2) – f(a(2))a'(2) = 17 * 4 – 5 * 1 = 68 – 5 = 63.

Our derivative of an integral calculator would ask for f(t) = "t*t + 1", b(x) = "x*x", a(x) = "x", b'(x) = "2*x", a'(x) = "1", and x = "2" to get this result.

Example 2: Trigonometric Functions

Find the derivative of F(x) = ∫_sin(x)^cos(x) (t) dt at x = π/4.

• f(t) = t
• b(x) = cos(x) => b'(x) = -sin(x)
• a(x) = sin(x) => a'(x) = cos(x)
• x = π/4 (approx 0.7854)

At x=π/4:

• b(π/4) = cos(π/4) = √2/2 ≈ 0.7071
• a(π/4) = sin(π/4) = √2/2 ≈ 0.7071
• f(b(π/4)) = √2/2
• f(a(π/4)) = √2/2
• b'(π/4) = -sin(π/4) = -√2/2
• a'(π/4) = cos(π/4) = √2/2

Derivative = (√2/2)*(-√2/2) – (√2/2)*(√2/2) = -2/4 – 2/4 = -1/2 – 1/2 = -1.

Using the derivative of an integral calculator with f(t)="t", b(x)="Math.cos(x)", a(x)="Math.sin(x)", b'(x)="-Math.sin(x)", a'(x)="Math.cos(x)", x="0.785398″ would yield -1.

How to Use This Derivative of an Integral Calculator

Using our derivative of an integral calculator is straightforward:

1. Enter the Integrand f(t): Type the function f(t) into the "Integrand f(t)" field. Use 't' as the variable and standard JavaScript math functions (e.g., `t*t`, `Math.sin(t)`, `Math.exp(t)`).
2. Enter the Upper Limit b(x): Input the expression for the upper limit b(x) in the "Upper Limit b(x)" field, using 'x' as the variable (e.g., `x*x`, `Math.cos(x)`).
3. Enter the Lower Limit a(x): Input the expression for the lower limit a(x) in the "Lower Limit a(x)" field, using 'x' as the variable (e.g., `x`, `2*x+1`).
4. Enter the Derivative b'(x): Manually calculate or look up the derivative of b(x) with respect to x, and enter it in the "Derivative b'(x)" field (e.g., `2*x`, `-Math.sin(x)`).
5. Enter the Derivative a'(x): Manually calculate or look up the derivative of a(x) with respect to x, and enter it in the "Derivative a'(x)" field (e.g., `1`, `2`).
6. Enter the Value of x: Input the specific numeric value of x at which you want to evaluate the derivative of the integral.
7. Calculate: The calculator will update automatically, or you can click "Calculate". The results, including the primary result (the derivative at x) and intermediate values, will be displayed. The table and chart will also update.
8. Read Results: The "Primary Result" shows the final derivative value. "Intermediate Results" and the table show values like b(x), a(x), f(b(x)), f(a(x)), b'(x), and a'(x) evaluated at the given x.
9. Reset: Click "Reset" to clear the fields to their default values.
10. Copy Results: Click "Copy Results" to copy the main result and intermediate steps to your clipboard.

The derivative of an integral calculator provides a quick way to check your manual calculations or to find the derivative when the expressions are complex. See also our integral calculator for definite integrals.

Key Factors That Affect Derivative of an Integral Results

The final value from the derivative of an integral calculator depends on several factors:

• The Integrand f(t): The form of f(t) directly influences f(b(x)) and f(a(x)). More complex integrands lead to more complex output expressions or values.
• The Upper Limit b(x): The function b(x) determines the upper point of evaluation for f and its derivative b'(x) contributes directly to the first term.
• The Lower Limit a(x): Similarly, a(x) determines the lower point and a'(x) contributes to the second term.
• The Derivatives b'(x) and a'(x): The rates of change of the limits are crucial multipliers in the formula. If the limits are constant, their derivatives are zero, simplifying the formula.
• The Value of x: The specific point x at which the derivative is evaluated determines the numerical values of a(x), b(x), a'(x), b'(x), and subsequently f(a(x)) and f(b(x)).
• Continuity and Differentiability: The formula assumes f(t) is continuous over the range [a(x), b(x)] (or [b(x), a(x)]) and that a(x) and b(x) are differentiable at the given x.

Frequently Asked Questions (FAQ)

Q: What is the formula used by the derivative of an integral calculator? A: The calculator uses the Leibniz integral rule: d/dx ∫_a(x)^b(x) f(t) dt = f(b(x))b'(x) – f(a(x))a'(x).

Q: What if the lower limit is constant and the upper limit is x? A: If a(x) = a (constant) and b(x) = x, then a'(x) = 0 and b'(x) = 1. The formula simplifies to f(x) * 1 – f(a) * 0 = f(x), which is the Fundamental Theorem of Calculus Part 1. Our derivative of an integral calculator handles this if you input a constant for a(x) and 0 for a'(x).

Q: What if the upper limit is constant and the lower limit is x? A: If b(x) = b (constant) and a(x) = x, then b'(x) = 0 and a'(x) = 1. The formula becomes 0 – f(x) * 1 = -f(x).

Q: Can I use this calculator if the limits are just constants? A: Yes. If a(x)=a and b(x)=b (constants), then a'(x)=0 and b'(x)=0, so the derivative of the integral (which is a constant value) is 0. The derivative of an integral calculator will show this.

Q: Why do I need to enter b'(x) and a'(x) myself? A: Implementing a full symbolic differentiator for arbitrary functions b(x) and a(x) within client-side JavaScript is very complex. The calculator requires you to provide these derivatives to ensure accuracy for a wide range of functions.

Q: What kind of expressions can I enter for f(t), a(x), and b(x)? A: You can enter standard JavaScript mathematical expressions, using 't' as the variable for f(t) and 'x' for a(x) and b(x). Examples: `t*t`, `Math.sin(t)`, `Math.exp(t*2)`, `x+5`, `x*x*x`.

Q: What happens if f(t), a(x), or b(x) are not continuous or differentiable? A: The formula and this derivative of an integral calculator assume f is continuous and a(x), b(x) are differentiable. If these conditions are not met at the point x, the result may not be valid.

Q: Can the calculator find the derivative as a function of x, not just at a point? A: This specific calculator evaluates the derivative at a given numeric value of x. To get the derivative as a function of x, you would apply the formula symbolically: f(b(x))b'(x) – f(a(x))a'(x), substituting the expressions for f, a, b, a', and b'. You can use our derivative calculator for simple functions.

Related Tools and Internal Resources

• Definite Integral Calculator: Calculate the value of definite integrals.
• Derivative Calculator: Find the derivative of various functions.
• Limit Calculator: Evaluate limits of functions.
• Function Plotter: Graph functions to visualize them.
• Series Calculator: Work with mathematical series.
• Equation Solver: Solve various types of equations.