Derivatives and the Shape of the Graph

Differentiation is a method to compute the rate at which a dependent output $y$ changes with respect to the change in the independent input $x$. This rate of change is called the derivative of $y$ with respect to $x$. In more precise language, the dependence of $y$ upon $x$ means that $y$ is a function of $x$. This functional relationship is often denoted $y=f(x)$, where $f$ denotes the function.

If $x$ and y are real numbers, and if the graph of $y$ is plotted against $x$, the derivative measures the slope of this graph at each point.

The simplest case is when $y$ is a linear function of $x$, meaning that the graph of $y$ divided by $x$ is a straight line. In this case, $y=f(x) = m \cdot x+b$, for real numbers $m$ and $b$, and the slope m is given by $\frac{\Delta y}{\Delta x}$, where the symbol $\Delta$ (the uppercase form of the Greek letter Delta) is an abbreviation for "change in. "

This formula is true because:

$y + \Delta y $

$= f(x+ \Delta x) $

$= m (x + \Delta x) + b $

$= m (x + \Delta x) + b $

$= y + m \Delta x$

It follows that:

$\Delta y = m \Delta x$

This gives an exact value for the slope of a straight line. If the function $f$ is not linear (i.e. its graph is not a straight line), however, then the change in $y$ divided by the change in $x$ varies: differentiation is a method to find an exact value for this rate of change at any given value of $x$.

Inflection Point

A point where the second derivative of a function changes sign is called an inflection point. At an inflection point, the second derivative may be zero, as in the case of the inflection point $x=0$ of the function $y=x^3$, or it may fail to exist, as in the case of the inflection point $x=0$ of the function $y=x^{\frac{1}{3}}$. At an inflection point, a function switches from being a convex function to being a concave function or vice versa.

Derivative

At each point, the derivative of is the slope of a line that is tangent to the curve. The line is always tangent to the blue curve; its slope is the derivative. Note derivative is positive where a green line appears, negative where a red line appears, and zero where a black line appears.