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How did Leibniz use infinitesimals to explain derivatives and integrals?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

Gottfried Wilhelm Leibniz developed his calculus by treating infinitesimals—quantities that are infinitely small yet not zero—as the foundation for his differential and integral calculus.12

Derivatives

Leibniz defined the derivative as a quotient of infinitesimals. When a variable $x$ undergoes an infinitesimal increase denoted as $d x$ , a function $f (x)$ experiences a corresponding infinitesimal change $d f$ . He expressed the derivative as the ratio:
$f^{'} (x) = \frac{d f}{d x}$
Leibniz viewed $d f$ as the difference between values of the function at two points separated by the infinitesimal distance $d x$ .2

Integrals

Leibniz conceptualized the integral as a summation of infinitesimals, using the elongated "S" symbol ( $\int$ ) to represent "sum." He viewed the integral $\int f (x) d x$ as the total area under a curve, calculated by summing the areas of infinitely many infinitesimally thin rectangles of width $d x$ .23

By using these concepts, Leibniz established the fundamental theorem of calculus as the realization that differentiation and integration are inverse operations, effectively treating the difference between successive sums as the final term in the sequence.2