In the previous lesson, we have reviewed the concepts and the handling of exponential and logarithmic functions. It is essential to develop a strong understanding of the basic rules and laws governing such functions’ analysis before attempting to try to understand its derivative.
Just as algebraic functions, differentiating exponential and logarithmic functions have its own set of rules. For the following formulas, let u be a function of x; a as a constant; and e is the Euler’s number.
Logarithmic Funcion to Base a:d(logau)=duulnaNatural Logarithmic Function:d(lnu)=duuExponential Function, eu:d(eu)=euduExponential Function, au:d(au)=aulnadu
Example 1: Find the derivative of the function y=log5(2x+3)
Solution:
Let:a=5u=2x+3du=2dx
To solve for the derivative of the function, use Equation1,
d(y)=d(log5(2x+3))dy=2dx(2x+3)ln5d(logau)=duulnadydx=2(2x+3)ln5Example 2: Find the derivative of the function y=loge(5−2x)
Solution:
Let:a=eu=5−2xdu=−2dx
To solve for the derivative of the function, use Equation1,
d(y)=d(loge(5−2x))dy=−2dx(5−2x)lneBut form the Properties of Natural Logarithms, lne=1. Thus,
dydx=−2(5−2x)(1)=−25−2xExample 3: Find the derivative of the function y=ln(5−2x)
Solution:
Let:u=5−2xdu=−2dx
To solve for the derivative of the function, use Equation2,
d(y)=d(ln(5−2x))dy=−2dx5−2xdydx=−25−2xAs you may have noticed, the answer is the same as the previous example. With this, we can verify that logeu is equal to lnu.
Example 4: Differentiate the function y=ln√2x2−5x+3 with respect to x.
Solution:
Let:u=√2x2−5x+3du=d(√2x2−5x+3)=d((2x2−5x+3)1/2)=12(2x2−5x+3)−1/2⋅d(2x2−5x+3)=12√2x2−5x+3⋅(4x−5)dx=4x−52√2x2−5x+3dx
To solve for the derivative of the function, use Equation2,
d(y)=d(ln√2x2−5x+3)dy=4x−52√2x2−5x+3dx√2x2−5x+3dy=4x−52√2x2−5x+3⋅√2x2−5x+3dxdydx=4x−52(√2x2−5x+3)2=4x−52(2x2−5x+3)Example 5: Determine the differential of the function y=5e3x.
Solution:
To solve for the differential of the function, use Equation3,
d(y)=d(5e3x)dy=5e3x⋅d(3x)=5e3x⋅3dx=15e3xdxExample 6: Find the derivative of the function with respect to t: x=e3te2t2.
Solution:
To solve for the derivative of the function, use Equation3,
d(x)=d(e3te2t2)dx=d(e3t+2t2)am+an=am+n=e3t+2t2⋅d(3t+2t2)=e3t+2t2⋅(3+4t)dtdxdt=(3+4t)e3t+2t2Example 7: Determine the differential of the function w=125(53v).
Solution:
To solve for the differential of the function, use Equation4,
d(w)=d(125(53v))dw=d(53⋅53v)=d(53+3v)=53+3vln5⋅d(3+3v)=53+3vln5⋅3dv=3ln5⋅53+3vdvExample 8: Find the derivative of y with respect to x of the function e12y=x2−1.
Solution:
In this example, the given equation is expressed as an implicit function. Hence, to determine its derivative, we can differentiate the given equation explicitly. d(e12y)=d(x2−1)e12y⋅d(12y)=2xdxey/2⋅12dy=2xdxdydx=4xey/2