The study of the definition, properties, and applications of the derivative of a function.

Derive is the geometry term.
Differentiate is the calculus term.
They mean the same thing.

to derive (geometry) == to differentiate (calculus) == to find the derivative
Derivation (geometry) == Differentiation (calculus) == the process of finding the derivative
Derivative (geometry) == differential equation (calculus).
Differential equations are equations involving derivatives.

differentiable equation == a function whose derivative exists at each point in its domain. It is smooth; it does not contain any break, angle or cusp.

The derivative is, in geometry, the slope of the tangent line of the function at the point of tangency.

Here is an animation of the tangent line along a curved line.

Here is a similar animation on youtube. https://www.youtube.com/watch?v=jGJdd0kzZDM

$$ \begin{aligned} \text{when }y &= f(x)\\ \\ \text{slope } &= \frac{\text{change in x}} {\text{change in y}} = \frac{dx}{dy} = \frac{\Delta x}{\Delta y} = f\prime(x) \end{aligned}$$

The “dx over dy” notation was invented by Gottfried Leibniz around 1700.

The “f prime” notation was by Lagrange.

How to Derive a Derivative

$y$ is a function of $x$. $y = ƒ(x)$

Therefore, an $(x,y)$ point on a graph can also be written as $(x,ƒ(x))$.
The graph below shows two points $(x,ƒ(x))$ and $((x+h),ƒ(x+h))$.

Given the two points $(x,ƒ(x))$ and $((x+h),ƒ(x+h))$, the slope of the secant line between the two points is

$$ \frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{(x+h)-(x)} = \frac{f(x+h)-f(x)}{h}$$

Then the slope at point $(x,ƒ(x))$ is given by the equation above where $h$ approaches zero, or

$$ \frac{\Delta f(x)}{\Delta x} = \frac{f(x+h)-f(x)}{(x+h)-(x)} = \frac{f(x+h)-f(x)}{h}$$

Many popular equations and their differentials can be found in the Gallery of Derivatives.

The normal distribution is given by the equation $$e^{\frac{-x^2}{2}}$$

The first derivative is $$f\prime(x) = xe^{\frac{-x^2}{2}} $$ The first derivative is odd. Asymmetric around the y axis.

The second derivative is $$f\prime\prime(x) = (x^2-1)e^{\frac{-x^2}{2}} $$

Distance and speed
Height and slope
Df/dx
Dy/dx
The straight line distance is 40 miles per hour 80 Mi after 2 hours.
Height of a person overtime the rate of growth over time the rate of change.
Wealth net worth vs saving or spending. Compounding interest is an example of exponential growth

The derivative allows you to find the minimum and the maximum points of the speed.

The second derivative tells you the bend of the first line weather it's concave or convex. At points where the bend changes the second derivative crosses the Y origin and this point is called the inflection point.

At a maximum the slope is 0 and the curve is bending down Words which means convex and the second derivative will be positive and at a minimum the slope again is zero and the band is upwards and the second derivative is negative or Versa or vice versa I'm not sure

The Liebniz notation for derivative is dy over dx. Sometimes df over dx.

Nowadays, the notation for partial derivative is

For a function with multiple variables, we do the derivative for only one variable, holding the others constant.

Notation. dy over dx is replaced with squiggle over squiggle x. Or y or z, whichever variable we're allowing to move.

“Partial derivative of blah with respect to x.”

Gradient