Derivative of exponential functions

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  1. 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}}}}}
    1. ddx  ex\frac{{d}}{{{d}x}}\;{{e}^x}
      ddx  esinx\frac{{d}}{{{d}x}}\;{{e}^{\sin x}}
      1. Differentiate:
        y=tan  (  cose5x2  )y = {tan\;}(\;\cos {{e}^{5{x^2}}}\;)
        1. ddx  x5\frac{d}{{dx}}\;{x^5} VS. ddx  5x\frac{d}{{dx}}\;{5^x}
          Topic Notes
          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)