Leibnitz rule
Understanding Leibnitz's Rule
Leibnitz's rule is a fundamental theorem in calculus that provides a formula for the differentiation of an integral whose limits are functions of the variable of differentiation. This rule is particularly useful when dealing with definite integrals where the limits are not constants but rather functions of the variable.
Leibnitz's Rule Formula
The Leibnitz rule can be stated as follows:
If $f(x, t)$ is continuous in $t$ and differentiable with respect to $x$, and if $\frac{\partial f}{\partial x}$ is continuous in both $x$ and $t$, then for two functions $a(x)$ and $b(x)$ which are differentiable, the following holds:
$$ \frac{d}{dx} \int_{a(x)}^{b(x)} f(x, t) \, dt = f(x, b(x)) \cdot b'(x) - f(x, a(x)) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x} \, dt $$
Breaking Down the Formula
The Leibnitz rule can be broken down into three distinct parts:
- The first term, $f(x, b(x)) \cdot b'(x)$, represents the impact of the upper limit's rate of change.
- The second term, $-f(x, a(x)) \cdot a'(x)$, represents the impact of the lower limit's rate of change.
- The third term, $\int_{a(x)}^{b(x)} \frac{\partial f}{\partial x} \, dt$, accounts for the differentiation of the integrand itself with respect to $x$.
Table of Important Points
| Point | Description |
|---|---|
| Continuity of $f$ | $f(x, t)$ must be continuous in $t$ for the rule to apply. |
| Differentiability in $x$ | $f(x, t)$ must be differentiable with respect to $x$. |
| Continuity of $\frac{\partial f}{\partial x}$ | The partial derivative of $f$ with respect to $x$ must be continuous in both $x$ and $t$. |
| Differentiable Limits | The limits $a(x)$ and $b(x)$ must be differentiable functions of $x$. |
Examples
Example 1: Basic Application
Let's consider the integral:
$$ I(x) = \int_{0}^{x^2} e^{xt} \, dt $$
To differentiate $I(x)$ with respect to $x$, we apply Leibnitz's rule:
$$ \frac{dI}{dx} = e^{x \cdot x^2} \cdot (x^2)' - e^{x \cdot 0} \cdot (0)' + \int_{0}^{x^2} \frac{\partial}{\partial x} e^{xt} \, dt $$
Simplifying, we get:
$$ \frac{dI}{dx} = e^{x^3} \cdot 2x + \int_{0}^{x^2} te^{xt} \, dt $$
Example 2: Variable Limits
Consider the integral:
$$ J(x) = \int_{\sin x}^{\cos x} \ln(1 + xt) \, dt $$
Differentiating $J(x)$ using Leibnitz's rule gives us:
$$ \frac{dJ}{dx} = \ln(1 + x \cos x) \cdot (-\sin x) - \ln(1 + x \sin x) \cdot \cos x + \int_{\sin x}^{\cos x} \frac{t}{1 + xt} \, dt $$
Conclusion
Leibnitz's rule is a powerful tool in calculus that allows us to differentiate under the integral sign when the limits of integration are functions of the variable of differentiation. It is essential to ensure that the functions involved meet the continuity and differentiability conditions required for the rule to be valid. Through practice and application of examples, one can gain proficiency in using Leibnitz's rule for solving complex problems in calculus.