Derivatives Basics

**What does derivative even mean?**

Good question, and there are multiple answers, but for our purposes we will define the derivative as the instantaneous rate of change of a function. Graphically, this is the slope of the function at a given point.

The derivative, like everything in calculus, is built upon limits. The formal definition of a derivative at a point is as follows:

$$f'(x)=\lim\limits_{x \to c}{\frac{f(x)-f(c)}{x-c}}$$

**Some terminology and basic facts to know:**

The commands "differentiate" and "derive" mean "find the derivative"

The process of taking the derivative is called "differentiation"

The derivative of a function $f(x)$ can be represented by the function $f'(x)$ (read "f prime of x")

Conversely, $f'(c)$ is the derivative at point $c$

The evaluation bar $\big\rvert$ tells you to evaluate the function at that point (i.e. $f(x)\big\rvert_3=f(3)$)

The derivative of a function can also be represented in Leibniz notation : $\frac{dy}{dx}$ (which means the derivative of function $y$ with respect to variable $x$). The "with respect to" part is important because it tells you which variable you are working on. This quantity is called the differential, and it represents a minute, almost negligible change in the function. As we will see later, the differential acts both as an operator and a value. Most of the time we take the derivative with respect to $x$, however this is not always the case. Leibniz notation can also be treated as a regular quotient: this is why the derivative of $x$ with respect to $x$ is $1$ ($\frac{dx}{dx}=1$)

Most functions can be differentiated more than once

** A bit about continuity:**

For a function to be differentiable at a point, it must be continuous at that point

Because you can't derive a function if it doesn't exist, we say that if a function is differentiable over an interval, it is also continuous on that interval; in other words, differentiability implies continuity

However, this does not work both ways; if a function is continuous, it does not *have* to be differentiable

The most prominent example of this is a cusp; cusps are not differentiable

Endpoints of functions are not differentiable since you cannot take the limit from both the left and right hand sides

Derivatives Formulas

Without proof, these are the four most common formulas that you will use to differentiate functions

1. Power Rule: used with polynomial expressions (functions with constant powers) $$f(x)=ax^n \to f'(x)= anx^{n-1}$$
ex. $\hspace{10mm}\frac{d}{dx}x^5=5x^4 \hspace{20mm}$ $\frac{d}{dx}7x^7=49x^6\hspace{20mm}$ $\frac{d}{dx}x^{\frac{1}{2}}=\frac{1}{2}x^{-\frac{1}{2}}$

2. Chain Rule: used with composite functions $$\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$$ Can be used with more than two composite functions i.e. $f(g(h(i(j(k(l(x)))))))$

ex. $\hspace{10mm}\frac{d}{dx} \sin(\cos(x^2))=\cos(\cos(x^2))\cdot -\sin(x^2)\cdot 2x$

3. Product Rule: used with product of two functions $$\frac{d}{dx} f(x)\cdot g(x)=f'(x)g(x)+g'(x)f(x)$$ ex. $\hspace{10mm}\frac{d}{dx} \sin(x)\cos(x)=\cos(x)\cdot \cos(x)+ -\sin(x) \cdot \sin(x) =$ $\cos^2(x)-\sin^2(x)$

4. Quotient Rule: used with quotient of two functions $$\frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)g(x)-g'(x)f(x)}{g(x)^2}$$ ex. $\hspace{10mm}\frac{d}{dx} \frac{x^3}{\sin(x)}=\frac{3x^2\sin(x)-\cos(x)x^3}{\sin^2(x)}$

2. Chain Rule: used with composite functions $$\frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)$$ Can be used with more than two composite functions i.e. $f(g(h(i(j(k(l(x)))))))$

ex. $\hspace{10mm}\frac{d}{dx} \sin(\cos(x^2))=\cos(\cos(x^2))\cdot -\sin(x^2)\cdot 2x$

3. Product Rule: used with product of two functions $$\frac{d}{dx} f(x)\cdot g(x)=f'(x)g(x)+g'(x)f(x)$$ ex. $\hspace{10mm}\frac{d}{dx} \sin(x)\cos(x)=\cos(x)\cdot \cos(x)+ -\sin(x) \cdot \sin(x) =$ $\cos^2(x)-\sin^2(x)$

4. Quotient Rule: used with quotient of two functions $$\frac{d}{dx} \frac{f(x)}{g(x)}=\frac{f'(x)g(x)-g'(x)f(x)}{g(x)^2}$$ ex. $\hspace{10mm}\frac{d}{dx} \frac{x^3}{\sin(x)}=\frac{3x^2\sin(x)-\cos(x)x^3}{\sin^2(x)}$

Common Derivatives

Here are some of the derivatives of common functions that will be helpful to remember (no proof)

$\frac{d}{dx} \sin(x)=\cos(x)$

$\frac{d}{dx} \cos(x)=-\sin (x)$

$\frac{d}{dx} e^x=e^x$

$\frac{d}{dx} \ln(x) = \frac{1}{x}$

$\frac{d}{dx} \cos(x)=-\sin (x)$

$\frac{d}{dx} e^x=e^x$

$\frac{d}{dx} \ln(x) = \frac{1}{x}$

Don't worry, this list will greatly expand in the next section

Tangent Lines

Lines of tangency, or tangent lines, are lines which have the slope of the graph at a certain point

The slope of these lines is the derivative of the function evaluated at that point

Equation of a tangent line at a point $c$:

$$y=f'(c)(x-c)+f(c)$$

This is just a reorganization of the point-slope line formula $(y-y_1)=m(x-x_1)$, with $m=f'(c)$

Normal lines run perpendicular to tangent lines at the point of tangency, meaning their slope is the negative reciprocal of that of the tangent line