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=fx with respect to x is represented as dydx or f'x or y' or y1.
  • 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=fx at a point Px, fx is defined as

limx0fx+x-fxx, if the limit exists at the point x, fx as shown in the figure.

 

Differentiation Rules

  • Derivative of a Constant: If c is any constant, then ddxc=0.

Example: ddx12=0

  • Derivative of (variable)constant: ddxxn=nxn-1, where n is a real number.

Example: ddxx4=4x4-1=4x3.

  • Derivative of Sum or Difference of functions: ddxfx±gx=ddxfx±ddxgx

Example: ddx12x2+10x4

=ddx12x2+ddx10x4

=122x+104x3

=24x+40x3

  • Derivative of Quotient of two functions: ddxfxgx=gxddxfx-fxddxgxgx2

Example: ddxx+1x2

=x2ddxx+1-x+1ddxx2x22

=-x-2x3

  • Derivative of Product of two functions: ddxfx·gx=fxddxgx+gxddxfx

Example: ddx3x2+1x2-1

=3x2+1ddxx2-1+x2-1ddx3x2+1

=3x2+12x+x2-16x

=12x3-4x

  • Derivative of the function of a function (Chain Rule): ddxfgx=ddxfgx·ddxgx

Example: ddx3x-43

=33x-43-1ddx3x-4

=33x-423

=93x-42

  • Derivative of Exponential Functions: ddxex=ex and ddxax=ax·ln a, where a>0 and a1.

Example: ddxex+2x

=ex+2x·ln 2

  • Derivative of Logarithmic Functions: ddxln x=1x and ddxlogax=1log a·x, where a>0 and a1.

Example: ddxln x+log3 x

=1x+1log 3·x

  • Derivatives of Trigonometric Functions:
    • ddxsin x=cos x
    • ddxcos x=-sin x
    • ddxtan x=sec2 x
    • ddxcot x=-cosec2 x
    • ddxsec x=sec x tan x
    • ddxcsc x=-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=fx is mentioned below.
    • First Derivative: dydx=f'x
    • Second Derivative: d2ydx2=ddxdydx=f''x
    • nth derivative: dnydxn=ddxdn-1ydxn-1=fnx

Example.: The first and second derivatives of a function y=x5+4x4-3x3 are:

First derivative:

y'=5x5-1+4·4x4-1-3·3x3-1

=5x4+16x3-9x2

Second derivative:

y''=5·4x4-1+16·3x3-1-9·2x2-1

=20x3+48x2-18x

Solved Examples

Example 1. Find the derivative of fx=x2+5x.

Solution: fx=x2+5x

f'x=2x+5

Example 2: Find dydx if y=tan 2x.

Solution: y=tan 2x

dydx=sec2 2x·2=2sec2 2x

Example 3: Find y' if y=2x+55x2-1.

Solution: y=2x+55x2-1

y'=2x+510x+25x2-1

=20x2+50x+10x2-2

=20x2+50x+10x2-2

=30x2+50x-2

Example 4: If y=9x-3, find y' at 1, 3.

Solution: y=9x-3

y'=0x-3-19x-32

=-9x-32

At -1, 3y'=-91-32=-9-22=-94

Example 5: Find the slope of the tangent line to the curve y=x2-4 at the point 3, 1.

Solution: The slope of the tangent line to a curve is its derivative. So, the slope of the tangent line to the curve y=x2-4 is

y'=2x

At 3, 1y'=2×3=6

Example 6: Find y' if y=17x.

Solution: y=17x

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

y'=17x·ln 17·12x-12

=17x·ln 172x

Example 7: Find f'x if fx=ln cot x.

Solution: fx=ln cot x

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

f'x=1cot x·-csc x cot x

=-csc x

Cheat Sheet

  • The derivative f'x of a function fx can be mathematically represented as f'x=limx0fx+x-fxx.
  • 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.
    • ddxconstant=0
    • ddxxn=nxn-1
    • ddxex=ex
    • ddxax=ax·ln a, where a>0 and a1
    • ddxln x=1x
    • ddxloga x=1ln a·x, where a>0 and a1
    • ddxfx±gx=ddxfx±ddxgx
    • ddxfx·gx=fx·ddxgx+gx·ddxfx
    • ddxfxgx=gx·ddxfx-fx·ddxgxgx2
    • ddxfgx=ddxfgx·ddxgx (Chain Rule)
    • ddxsin x=cos x
    • ddxcos x=-sin x
    • ddxtan x=sec2 x
    • ddxcot x=-cosec2 x
    • ddxsec x=sec x tan x
    • ddxcsc x=-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.