Index
Limits
Fundamental Theorem of Calculus
Derivatives
Integrals
Leibniz Integral Rule
The Leibniz Integral rule governs how to take the derivative of a definite integral wrt $x$ when the the integral itself is wrt some other variable $y$ and, but the both the integrand $f(x,y)$ and the bounds of the integral $a(x)$ and $b(x)$ vary in $x$. It is an advanced calculus concept stated below.
\[\frac{d}{dx} \left( \int^{b(x)}_{a(x)} f(x,y) dy \right)\]
\[= f(x, b(x)) \cdot \frac{d}{dx} b(x) - f(x, a(x)) \cdot \frac{d}{dx}a(x) + \int^{b(x)}_{a(x)} \frac{\partial}{\partial x} f(x,y) dy\]
The 3 terms in this rule are intuitively explained as follows.
- Term 1 covers how the integral changes with changes to the upper integral bound (notice that it is + because increasing values of b(x) extend the range of the integral)
- Term 2 covers how the integral changes with changes to the lower integral bound (notice that it is - because decreasing values of a(x) reduce the range of the integral)
- The last term should be familiar from the fundamental theorem of calculus.
The Beauty of $e$
Definition of $e$
Infinite Series Definition
Calculus-Based Definition
What this means
Exponential Growth and Decay
Logistic Growth and Decay