Derivative of exponential and logarithmic functions

1 Mathematics Learning CentreDerivatives of exponential andlogarithmic functionsChristopher Thomasc 1997 University of SydneyMathematics Learning Centre, University of Sydney11 derivatives of exponential and logarithmic func-tionsIf you are not familiar with exponential and logarithmic functions you may wish to consultthe bookletExponents and Logarithmswhich is available from the Mathematics seen that there are two notations popularly used for natural logarithms,logeand ln. These are just two different ways of writing exactly the same thing, so thatlogex booklet we will use both these basic results are:ddxex=exddx(logex)= use these results and the rules that we have learnt already to differentiate functionswhich involve exponentials or loge(x2+3x+1).SolutionWesolve this by using the chain rule and our knowledge of the Derivative of (x2+3x+1) =ddx(logeu)(whereu=x2+3x+1)=ddu(logeu) dudx(by the chain rule)=1u dudx=1x2+3x+1 ddx(x2+3x+1)=1x2+3x+1 (2x+3)=2x+3x2+3x+ (e3x2).Mathematics Learning Centre, University of Sydney2 SolutionThis is an application of the chain rule together with our knowledge of the Derivative (e3x2)=deudxwhereu=3x2=deudu dudxbythe chain rule=eu dudx=e3x2 ddx(3x2)= (ex3+2x).

2 SolutionAgain, we use our knowledge of the Derivative ofextogether with the chain (ex3+2x)=deudx(whereu=x3+2x)=eu dudx(by the chain rule)=ex3+2x ddx(x3+2x)=(3x2+2) ex3+ ln (2x3+5x2 3).SolutionWesolve this by using the chain rule and our knowledge of the Derivative of (2x3+5x2 3) =dlnudx(whereu=(2x3+5x2 3)=dlnudu dudx(by the chain rule)=1u dudx=12x3+5x2 3 ddx(2x3+5x2 3)=12x3+5x2 3 (6x2+10x)=6x2+10x2x3+5x2 Learning Centre, University of Sydney3 There are two shortcuts to differentiating functions involving exponents and four examples above gaveddx(loge(x2+3x+1)) =2x+3x2+3x+1ddx(e3x2)=6xe3x2ddx(ex3+2x)= (3x2+2)e3x2ddx(loge(2x3+5x2 3)) =6x2+10x2x3+5x2 examples suggest the general rulesddx(ef(x))=f (x)ef(x)ddx(lnf(x)) =f (x)f(x).These rules arise from the chain rule and the fact thatdexdx=exanddlnxdx= canspeed up the process of differentiation but it is not necessary that you remember you forget, just use the chain rule as in the examples 1 Differentiate the following (x)=ln(2x3) (x)= (x)=ln(11x7) (x)=ex2+ (x)=loge(7x 2) (x)=e (x)=ln(ex+x3) (x)=ln(exx3) (x)=ln x2+1x3 x Mathematics Learning Centre, University of Sydney4 Solutions to Exercise (x)=6x22x3=3xAlternatively writef(x)=ln2+3lnxso thatf (x)= (x)= (x)= (x)=(2x+3x2)ex2+ (x)=loge7 2logexso thatf (x)= (x)= e (x)=ex+3x2ex+ (x)=lnex+3lnxso thatf (x)=1+ (x)=ln(x2+1) ln(x3 x)sothatf (x)=2xx2+1 3x2 1x3 x.)