Differentiation §

Definition: Differentiation is a fundamental concept in calculus that deals with the rate at which a function changes. The derivative of a function gives us the slope of the tangent line to the curve of that function at any given point.

For example, the graph below ($y=f(x)$) is a continuous functions on the interval $[a,x]$ (in this case $a=1,x=4$), so the average rate of change of $y$ with respect to $x$ in the interval $[a,x]$ is $\frac{\Delta y}{\Delta x}=\frac{f(x)-f(a)}{x-a}$. This is the difference quotient and is used to define the derivative - usually using the limit definition.

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f(x)=-((x-3)^2)+5

The limit definition of differentiation is foundational to understanding how the concept of the derivative is established in calculus. The derivative of a function at a point is defined as the limit of the average rate of change of the function over a small interval around that point as the interval’s width approaches zero.

Given a function $f(x)$, the average rate of change of $f$ over the interval $[x,x+h]$ is the difference quotient:

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

As $h$ approaches zero, this average rate of change approaches the instantaneous rate of change of $f$ at $x$. This limit, if it exists, is the derivative of $f$ at $x$.

Formally, the derivative of $f$ with respect to $x$, denoted $f^{\prime }(x)$ or $\frac{df}{dx}$, is given by:

$f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$

Provided the limit exists. In summation,

Formal Differentiability Definition §

Definition: Suppose that $f(x)$ is defined on an interval $(a,b)$ containing the point $x_0$ (i.e. $a<x_0<b$). Then, $f(x)$ is differentiable at $x_0$ if and only if the following limit exists: $f^{\prime }(x)={\lim }_{h\to 0}\frac{f(x+h)-f(x)}{h}$ $f(x)$ is a differentiable function if every point in its domain is differentiable by the above statement.

Intuitive Explanation §

Imagine you’re walking along a curved path and you want to know how steep the path is at a particular point. One way to figure it out is to look at how much higher or lower you’d be if you took a small step forward (that’s the difference in the function values) and divide it by the size of the step (that’s $h$). This gives you an average slope over that small step. As you take smaller and smaller steps (making $h$ approach 0), you get a better and better idea of the slope exactly at the point you’re standing on. The limit captures this idea: as $h$ gets infinitely small, the average slope becomes the instantaneous slope, which is the derivative.

Example §

To find the derivative of the function $f(x)=x^2$ at a general point $x$ using the limit definition:

$f^{\prime }(x)={\lim }_{h\to 0}\frac{(x+h{)}^2-x^2}{h}$

Expanding and simplifying:

$f^{\prime }(x)={\lim }_{h\to 0}\frac{x^2+2xh+h^2-x^2}{h}$ $={\lim }_{h\to 0}(2x+h)$ $=2x$

So, the derivative of $f(x)=x^2$ is $f^{\prime }(x)=2x$.

Notation: §

There are various ways to denote the derivative of a function, as shown above in practice, as well:

1. If $y=f(x)$, then the derivative is represented as $f^{\prime }(x)$ or $\frac{dy}{dx}$.
2. For higher-order derivatives: $f^{\prime \prime }(x)$, $f^{\prime \prime \prime }(x)$, and $f^{(n)}(x)$ for the nth derivative.
3. The differential operator, $D$, can also be used: $Df(x)$.

Basic Rules: §

1. Constant Rule: The derivative of a constant is zero.

   $\frac{d}{dx}c=0$

   where $c$ is a constant.

2. Power Rule: For any real number $n$,

   $\frac{d}{dx}{x}^n=n{x}^{n-1}$

3. Constant Multiple Rule: If $c$ is a constant and $f(x)$ is a function, then

   $\frac{d}{dx}[c\cdot f(x)]=c\cdot \frac{d}{dx}f(x)$

4. Sum Rule: If $f(x)$ and $g(x)$ are functions, then

   $\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$

5. Product Rule: For functions $f(x)$ and $g(x)$,

   $\frac{d}{dx}[f(x)\cdot g(x)]=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$

6. Quotient Rule: For functions $f(x)$ and $g(x)$ where $g(x)\ne 0$,

   $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$

Logarithmic Differentiation §

Related to implicit differentiation.

Begin by writing $y=f(x)$, where $f(x)$ is the function to be differentiated. We then take natural logarithms of both sides, and simplify $\ln (f(x))$ using the laws of logarithms. Then, differentiate both sides of the equation with respect to $x$, then solving the resulting equation for $\frac{dy}{dx}$.

Example: Differentiate $\frac{x^2\sin x}{\cos 2x}$.

$$y=\frac{x^2\sin x}{\cos 2x}⟹\ln y=\ln \left(\frac{x^2\sin x}{\cos 2x}\right)$$

$$⟹\ln y=2\ln x+\ln \sin x-\ln \cos 2x$$

$$⟹\frac{d}{dx}\left(\ln y\right)=\frac{d}{dx}\left(2\ln x+\ln \sin x-\ln \cos 2x\right)$$

$$⟹\frac{1}{y}\frac{dy}{dx}=\frac{2}{x}+\frac{\cos x}{\sin x}+2\frac{\sin 2x}{\cos 2x}$$

$$⟹\frac{dy}{dx}=\frac{2y}{x}+\cot x+2\tan 2x:y=\frac{x^2\sin x}{\cos 2x}$$

$$\therefore \frac{dy}{dx}=2\frac{x\sin x}{\cos 2x}+\cot x+2\tan 2x$$

Higher Order Derivatives §

• For functions that require differentiation multiple times, we use notation like $f^{\prime \prime }(x)$ or $f^{(iv)}(x)$ to denote the second, third, fourth, etc., derivatives respectively.

Basic Derivatives: §

1. $\frac{d}{dx}x=1$
2. $\frac{d}{dx}{x}^2=2x$
3. $\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt{x}}$
4. $\frac{d}{dx}\ln (x)=\frac{1}{x}$
5. $\frac{d}{dx}{e}^x=e^x$
6. $\frac{d}{dx}\sin (x)=\cos (x)$
7. $\frac{d}{dx}\cos (x)=-\sin (x)$

Applications: §

1. Tangents and Normals: The derivative gives the slope of the tangent to the curve at a particular point. Thus, knowing the point and the slope, the equation of the tangent can be found.
2. Motion: In physics, the derivative of position with respect to time gives velocity, and the second derivative gives acceleration.
3. Optimization Problems: Differentiation can be used to find maximum or minimum values of functions.

Challenges: §

1. Differentiate the function $y=3x^2-4x+7$.
2. Using the Product Rule, differentiate $y=x^2\sin (x)$.
3. Find the equation of the tangent line to the curve $y=x^2$ at the point $(1,1)$.

Remember, practice makes perfect! Regularly working on problems involving differentiation will help solidify your understanding.