Mobile Notice
It seems that you are on your mobile phone, which means that the screen's width is narrow for some equations. For the best viewing experience, you can view this website in landscape mode.

Derivatives of Exponential and Logarithmic Functions

Author | Engr. Nico O. Aspra, M.Eng., RMP, LPT

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)ln5

Example 2: Find the derivative of the function y=loge(52x)

Solution:
Let:a=eu=52xdu=2dx

To solve for the derivative of the function, use Equation1,

d(y)=d(loge(52x))dy=2dx(52x)lne

But form the Properties of Natural Logarithms, lne=1. Thus,

dydx=2(52x)(1)=252x

Example 3: Find the derivative of the function y=ln(52x)

Solution:
Let:u=52xdu=2dx

To solve for the derivative of the function, use Equation2,

d(y)=d(ln(52x))dy=2dx52xdydx=252x

As 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=ln2x25x+3 with respect to x.

Solution:
Let:u=2x25x+3du=d(2x25x+3)=d((2x25x+3)1/2)=12(2x25x+3)1/2d(2x25x+3)=122x25x+3(4x5)dx=4x522x25x+3dx

To solve for the derivative of the function, use Equation2,

d(y)=d(ln2x25x+3)dy=4x522x25x+3dx2x25x+3dy=4x522x25x+32x25x+3dxdydx=4x52(2x25x+3)2=4x52(2x25x+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=5e3xd(3x)=5e3x3dx=15e3xdx

Example 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+2t2d(3t+2t2)=e3t+2t2(3+4t)dtdxdt=(3+4t)e3t+2t2

Example 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(5353v)=d(53+3v)=53+3vln5d(3+3v)=53+3vln53dv=3ln553+3vdv

Example 8: Find the derivative of y with respect to x of the function e12y=x21.

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(x21)e12yd(12y)=2xdxey/212dy=2xdxdydx=4xey/2