Precalculus - Introduction to Derivatives

Introduction to Derivatives

• The instantaneous change of a function with respect to an independent variable is called the derivative of the function.
• The derivative of a function $y=f\left(x\right)$ with respect to $x$ is represented as $\frac{dy}{dx}$ or $f^{\text{'}}\left(x\right)$ or $y^{\text{'}}$ or $y^{\left(1\right)}$.
• Geometrically, the derivative of a function at a point is the slope of the tangent line on the curve of the function at that point.
• Mathematically, the derivative of a function $y=f\left(x\right)$ at a point $P\left(x, f\left(x\right)\right)$ is defined as

$\underset{∆x\to 0}{\lim }\frac{f\left(x+∆x\right)-f\left(x\right)}{∆x}$, if the limit exists at the point $\left(x, f\left(x\right)\right)$ as shown in the figure.

 

Differentiation Rules

• Derivative of a Constant: If $c$ is any constant, then $\frac{d}{dx}\left(c\right)=0$.

Example: $\frac{d}{dx}\left(12\right)=0$

• Derivative of ${\left(\text{variable}\right)}^{\mathbf{\text{constant}}}$: $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$, where $n$ is a real number.

Example: $\frac{d}{dx}\left(x^4\right)=4x^{4-1}=4x^3$.

• Derivative of Sum or Difference of functions: $\frac{d}{dx}\left(f\left(x\right)\pm g\left(x\right)\right)=\frac{d}{dx}f\left(x\right)\pm \frac{d}{dx}g\left(x\right)$

Example: $\frac{d}{dx}\left(12x^2+10x^4\right)$

$=\frac{d}{dx}\left(12x^2\right)+\frac{d}{dx}\left(10x^4\right)$

$=12\left(2x\right)+10\left(4x^3\right)$

$=24x+40x^3$

• Derivative of Quotient of two functions: $\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{g\left(x\right){\displaystyle \frac{d}{dx}}\left(f\left(x\right)\right)-f\left(x\right){\displaystyle \frac{d}{dx}}\left(g\left(x\right)\right)}{{\left(g\left(x\right)\right)}^2}$

Example: $\frac{d}{dx}\left(\frac{x+1}{x^2}\right)$

$=\frac{\left(x^2\right){\displaystyle \frac{d}{dx}}\left(x+1\right)-\left(x+1\right){\displaystyle \frac{d}{dx}}\left(x^2\right)}{{\left(x^2\right)}^2}$

$=\frac{-x-2}{x^3}$

• Derivative of Product of two functions: $\frac{d}{dx}f\left(x\right)·g\left(x\right)=f\left(x\right)\frac{d}{dx}g\left(x\right)+g\left(x\right)\frac{d}{dx}f\left(x\right)$

Example: $\frac{d}{dx}\left[\left(3x^2+1\right)\left(x^2-1\right)\right]$

$=\left(3x^2+1\right)\frac{d}{dx}\left(x^2-1\right)+$$\left(x^2-1\right)\frac{d}{dx}\left(3x^2+1\right)$

$=\left(3x^2+1\right)\left(2x\right)+\left(x^2-1\right)\left(6x\right)$

$=12x^3-4x$

• Derivative of the function of a function (Chain Rule): $\frac{d}{dx}f\left(g\left(x\right)\right)=\frac{d}{dx}f\left(g\left(x\right)\right)·\frac{d}{dx}g\left(x\right)$

Example: $\frac{d}{dx}{\left(3x-4\right)}^3$

$=3{\left(3x-4\right)}^{3-1}\frac{d}{dx}\left(3x-4\right)$

$=3{\left(3x-4\right)}^2\left(3\right)$

$=9{\left(3x-4\right)}^2$

• Derivative of Exponential Functions: $\frac{d}{dx}\left(e^x\right)=e^x$ and $\frac{d}{dx}\left(a^x\right)=a^x·\ln \left|a\right|$, where $a>0$ and $a\ne 1$.

Example: $\frac{d}{dx}\left(e^x+2^x\right)$

$=e^x+2^x·\ln \left|2\right|$

• Derivative of Logarithmic Functions: $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$ and $\frac{d}{dx}\left[{\log }_{a}x\right]=\frac{1}{\left(\log a\right)·x}$, where $a>0$ and $a\ne 1$.

Example: $\frac{d}{dx}\left(\ln x+{\log }_3 x\right)$

$=\frac{1}{x}+\frac{1}{\left(\log 3\right)·x}$

• Derivatives of Trigonometric Functions:

• • $\frac{d}{dx}\left(\sin x\right)=\cos x$
  • $\frac{d}{dx}\left(\cos x\right)=-\sin x$
  • $\frac{d}{dx}\left(\tan x\right)=se{c}^2 x$
  • $\frac{d}{dx}\left(cot x\right)=-\mathrm{co}se{c}^2 x$
  • $\frac{d}{dx}\left(sec x\right)=sec x \tan x$
  • $\frac{d}{dx}\left(csc x\right)=-csc x cot x$

• Higher Order Derivatives: Differentiating a function more than once (successive derivatives of a function) results in higher order derivatives. The higher-order derivatives of a function $y=f\left(x\right)$ is mentioned below.

• • First Derivative: $\frac{dy}{dx}=f^{\text{'}}\left(x\right)$
  • Second Derivative: $\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right)=f^{\text{'}\text{'}}\left(x\right)$
  • $n^{th}$ derivative: $\frac{d^{n}y}{d{x}^n}=\frac{d}{dx}\left(\frac{d^{n-1}y}{d{x}^{n-1}}\right)=f^n\left(x\right)$

Example.: The first and second derivatives of a function $y=x^5+4x^4-3x^3$ are:

First derivative:

$y^{\text{'}}=5x^{5-1}+4·4x^{4-1}-3·3x^{3-1}$

$=5x^4+16x^3-9x^2$

Second derivative:

$y^{\text{'}\text{'}}=5·4x^{4-1}+16·3x^{3-1}-9·2x^{2-1}$

$=20x^3+48x^2-18x$

Solved Examples

Example 1. Find the derivative of $f\left(x\right)=x^2+5x$.

Solution: $f\left(x\right)=x^2+5x$

$f^{\text{'}}\left(x\right)=2x+5$

Example 2: Find $\frac{dy}{dx}$ if $y=\tan 2x$.

Solution: $y=\tan 2x$

$\frac{dy}{dx}=\left(se{c}^2 2x\right)·2$$=2se{c}^2 2x$

Example 3: Find $y^{\text{'}}$ if $y=\left(2x+5\right)\left(5x^2-1\right)$.

Solution: $y=\left(2x+5\right)\left(5x^2-1\right)$

$y^{\text{'}}=\left(2x+5\right)\left(10x\right)+\left(2\right)\left(5x^2-1\right)$

$=\left(20x^2+50x\right)+\left(10x^2-2\right)$

$=20x^2+50x+10x^2-2$

$=30x^2+50x-2$

Example 4: If $y=\frac{9}{\left(x-3\right)}$, find $y^{\text{'}}$ at $\left(1, 3\right)$.

Solution: $y=\frac{9}{\left(x-3\right)}$

$y^{\text{'}}=\frac{0\left(x-3\right)-1\left(9\right)}{{\left(x-3\right)}^2}$

$=\frac{-9}{{\left(x-3\right)}^2}$

At $\left(-1, 3\right)$: $y^{\text{'}}=\frac{-9}{{\left(1-3\right)}^2}$$=\frac{-9}{{\left(-2\right)}^2}$$=-\frac{9}{4}$

Example 5: Find the slope of the tangent line to the curve $y=x^2-4$ at the point $\left(3, 1\right)$.

Solution: The slope of the tangent line to a curve is its derivative. So, the slope of the tangent line to the curve $y=x^2-4$ is

$y^{\text{'}}=2x$

At $\left(3, 1\right)$: $y^{\text{'}}=2\times 3$$=6$

Example 6: Find $y^{\text{'}}$ if $y={17}^{\sqrt{x}}$.

Solution: $y={17}^{\sqrt{x}}$

Let us apply the chain rule to directly find the derivative of the given function.

$y^{\text{'}}={17}^{\sqrt{x}}·\left(\ln 17\right)·\frac{1}{2}{x}^{-\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

$=\frac{{17}^{\sqrt{x}}·\left(\ln 17\right)}{2\sqrt{x}}$

Example 7: Find $f^{\text{'}}\left(x\right)$ if $f\left(x\right)=\ln \left(cot x\right)$.

Solution: $f\left(x\right)=\ln \left(cot x\right)$

Let us apply the chain rule to directly find the derivative of the given function.

$f^{\text{'}}\left(x\right)=\frac{1}{cot x}·\left(-csc x cot x\right)$

$=-csc x$

Cheat Sheet

• The derivative $f^{\text{'}}\left(x\right)$ of a function $f\left(x\right)$ can be mathematically represented as $f^{\text{'}}\left(x\right)=\underset{∆x\to 0}{\lim }\frac{f\left(x+∆x\right)-f\left(x\right)}{∆x}$.
• In simple terms, the derivative of a function at a point is simply the slope of the tangent line on the curve of the function at that point.
• Some of the standard results of differentiation are mentioned below.

• • $\frac{d}{dx}\left(\text{constant}\right)=0$
  • $\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$
  • $\frac{d}{dx}\left(e^x\right)=e^x$
  • $\frac{d}{dx}\left(a^x\right)=a^x·\ln \left|a\right|$, where $a>0$ and $a\ne 1$
  • $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$
  • $\frac{d}{dx}\left({\log }_a x\right)=\frac{1}{\left(\ln a\right)·x}$, where $a>0$ and $a\ne 1$
  • $\frac{d}{dx}\left[f\left(x\right)\pm g\left(x\right)\right]=\frac{d}{dx}\left[f\left(x\right)\right]\pm \frac{d}{dx}\left[g\left(x\right)\right]$
  • $\frac{d}{dx}\left[f\left(x\right)·g\left(x\right)\right]$$=f\left(x\right)·\frac{d}{dx}\left[g\left(x\right)\right]+g\left(x\right)·\frac{d}{dx}\left[f\left(x\right)\right]$
  • $\frac{d}{dx}\left[\frac{f\left(x\right)}{g\left(x\right)}\right]=\frac{g\left(x\right)·\frac{d}{dx}\left[f\left(x\right)\right]-f\left(x\right)·\frac{d}{dx}\left[g\left(x\right)\right]}{{\left[g\left(x\right)\right]}^2}$
  • $\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]=\frac{d}{dx}\left[f\left(g\left(x\right)\right)\right]·\frac{d}{dx}\left[g\left(x\right)\right]$ (Chain Rule)

• • $\frac{d}{dx}\left(\sin x\right)=\cos x$
  • $\frac{d}{dx}\left(\cos x\right)=-\sin x$
  • $\frac{d}{dx}\left(\tan x\right)=se{c}^2 x$
  • $\frac{d}{dx}\left(cot x\right)=-\mathrm{co}se{c}^2 x$
  • $\frac{d}{dx}\left(sec x\right)=sec x \tan x$
  • $\frac{d}{dx}\left(csc x\right)=-csc x cot x$

Blunder Areas

• Always be mindful that not all continuous functions are differentiable.
• It is helpful to examine the type of function based on its equation or graph to determine whether it is differentiable or not.
• Differentiation formulas, comprehensively discussed in other lessons, are helpful and quick ways to find the derivative of a function.