Derivative of exponential functions

Derivative of exponential functions

An exponential function is a function containing a numerical base with at least one variable in its exponent. In this section, we will learn how to differentiate exponential functions, including natural exponential functions and other composite functions that require the application of the Chain Rule.


Differential Rules – Exponential Functions

ddx  cx=cxlnc\frac{{d}}{{{d}x}}\;{c^x} = {c^x} \cdot \ln c
ddx  c(        )=c(        )lncddx(        )\frac{{d}}{{{d}x}}\;{c^{\left( {\;\;\;\;} \right)}} = {c^{\left( {\;\;\;\;} \right)}} \cdot \ln c \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)

ddx  ex=ex\frac{{d}}{{{d}x}}\;{{e}^x} = {{e}^x}
ddx  e(        )=e(        )ddx(        )\frac{{d}}{{{d}x}}\;{{e}^{\left( {\;\;\;\;} \right)}} = {{e}^{\left( {\;\;\;\;} \right)}} \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
  • Introduction
    ddx  2x\frac{d}{{dx}}\;{2^x}
    ddx  24x3\frac{d}{{dx}}\;{2^{4{x^3}}}

  • 1.
    ddx  35x2\frac{d}{{dx}}\;{3^{{5^{{x^2}}}}}

  • 2.
    ddx  ex\frac{{d}}{{{d}x}}\;{{e}^x}
    ddx  esinx\frac{{d}}{{{d}x}}\;{{e}^{\sin x}}

  • 3.
    y=tan  (  cose5x2  )y = {tan\;}(\;\cos {{e}^{5{x^2}}}\;)

  • 4.
    ddx  x5\frac{d}{{dx}}\;{x^5} VS. ddx  5x\frac{d}{{dx}}\;{5^x}