Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te sciences, weter in biology, pysics, matematics, cemistry, or anoter closely related field A common topic in calculus and iger matematics is te derivative It is my ope tat after reading tese instructions, you will not only understand wat a derivative is, but be capable of finding te derivative of any function witin five minutes Wat is a Derivative? We know from prior matematics courses tat te formula for te slope of te line connecting any two points on a grap, also known as te secant line, is y A secant line between or f(x 2 ) f(x 1) x x 2 x 1 two points tat are extremely close togeter is called a tangent line A derivative is te slope of te tangent line to a grap at a given point, and tis document will walk you troug te process of finding te derivative of a function Below, we see a secant line and a tangent line drawn on a curve Imagine tat te two points of intersection between te secant line and te curve above move closer and closer to eac oter In fact, imagine tat tey move witin 1 units of eac oter Let s call tis very small distance between te two points Now, picture a secant line being drawn between tese two points tat are really close togeter, separated by a distance of Tis secant line would be a good approximation of te tangent line to te curve at tis point Te slope of tis tangent line is called te derivative
How to Find te Derivative of a Function Now tat we understand te concept of a derivative, let s begin te process of finding te derivative of a function Work slowly troug tese steps, as a strong understanding of te concept of a derivative is very important for success in calculus and many upper level matematics courses 1 Find te slope formula for a derivative Remember tat te derivative is just te slope of te tangent line, and a tangent line is a secant line drawn between two points tat are very close togeter Call tese points x, f(x) and x +, f(x + ), were is an incredibly small decimal First, find te slope between tese two points: y = f(x+) f(x) = f(x+) f(x) x (x+) x It is not enoug to just say tat is small, owever State tat is a very small number in your formula by making use of te it To sow tat you want to be a small number using its, take te it of te above expression as approaces, and you are left wit te following: f(x+) f(x) Tis equation above is called te it definition of te derivative Soon, you will learn quicker ways to find te derivative of a function, but for now focus on ow to use tis definition 2 Evaluate f(x+) f(x) After finding te slope formula, find f(x+) f(x) Tis is no different tan finding f(2) Plug in x + everywere an x appears, just as you would plug in 2 wen findingf(2) After doing tis, subtract from f(x + ) te original function, denotedf(x), and ten divide tis difference by Example: Given f(x) = 5x 2 + 7, evaluate f(x+) f(x) f(x + ) f(x) = 5(x + )2 + 7 (5x + 7)
3 Use appropriate algebraic tecniques to simplify te numerator and denominator Now tat you ave found f(x+) f(x), simplify your numerator and denominator using algebra First, begin by using te FOIL metod* tat you learned in previous mat classes to simplify f(x + ) After doing tis, cancel terms on te left and side and rigt and side of te subtraction sign between f(x + )and f(x) [5(x 2 + 2x + 2 ) + 7 (5x 2 + 7) = 5x2 + 1x + 5 2 + 7 5x 2 7 = 5x2 + 1x + 5 2 + 7 5x 2 7 = 1x + 52 4 Take te it as approaces After you ave simplified f(x+) f(x), take te it of tis quotient as approaces f(x + ) f(x) 1x 5 2 After initially plugging in, you will receive te following output: 1x() 5() 2 N O T E *In case you ave forgotten, te FOIL metod stands for First, Outside, Inside, Last So, in te above problem, we ave te following calculation: (x + ) 2 = (x + ) (x + ) First, multiply te first (left most) terms witin eac set of parenteses, giving x x = x 2 Next, multiply te terms on te outside of eac set of parenteses, giving x = x Ten, multiply te terms on te inside of eac set of parenteses, giving x = x Finally, multiply te last (rigt most) terms witin eac set of parenteses, giving = 2
5 After receiving an output of te form, cancel as many terms as possible Wen using te it definition of te derivative, tis will always be te case Terefore, if is not received after initially taking te it, ceck your work as tere is a mistake An output of indicates tat tere are common terms in te numerator and denominator tat can be cancelled Wen taking a derivative, always seek to factor out** and cancel te terms 1x 5 2 (1x 5) (1x 5) 1x 5 6 After cancelling all possible terms, re-evaluate your it After you ave cancelled terms in te numerator wit terms in te denominator, reevaluate your it In te end, tere sould only be x terms remaining 1x 5 = 1x 5() = 1x Tus, te derivative of f(x) = 5x 2 + 7 is 1x Te derivative, 1x, is a function tat provides te slope of te tangent line to our curve, f(x), at any given x-value N O T E **In case you ave forgotten ow to factor polynomials, let s go troug te steps We are seeking to someow factor 1x 52 Always look on te left and rigt side of te subtraction or addition sign and tink Wat variables do I ave in common? We see tat on te left and rigt side of te subtraction sign we ave at least one, so factor out one Always factor out te common terms to te least power present Here, we ave an and an 2, so we factor out only one We may now write: 1x 5 2 (1x 5) =
Summary Tere are many different ways to take a derivative As you will soon learn, tis is te longest way possible Tere are, in fact, sortcuts to taking te derivative of almost any function Tese sortcuts, owever, will only make sense if te concept of a derivative is truly understood In conclusion, you want to remember te six steps to taking a derivative: 1 Write down te slope formula, namely f(x+) f(x) 2 Evaluate f(x+) f(x) 3 Use algebraic tecniques to simplify te result of f(x+) f(x) 4 Take te it as approaces of f(x+) f(x) 5 After receiving an output of te form, cancel as many terms as possible 6 After cancelling all possible terms, re-evaluate your it Examples Let s practice your derivative-taking skills Walk troug te example below on te left, and ten try te problem on te rigt on your own Find te derivative of f(x) = 4x 2 8 using te steps outlined above Find te derivative of f(x) = 9 3x 2 using te steps outline above 1 f(x+) f(x) 2 f(x+) f(x) = 4(x+)2 8 (4x 2 8) 1 2 3 4x2 +8x+4 2 8 4x 2 +8 = 8x+42 8x+4 4 2 8x()+4() 2 5 (8x+4) = 3 4 8x + 4 5 6 8x + 4 = 8x + 4() = 8x 6