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.

Lessons

Differential Rules – Exponential Functions

ddxcx=cx⋅lnc\frac{{d}}{{{d}x}}\;{c^x} = {c^x} \cdot \ln c
ddxc()=c()⋅lnc⋅ddx()\frac{{d}}{{{d}x}}\;{c^{\left( {\;\;\;\;} \right)}} = {c^{\left( {\;\;\;\;} \right)}} \cdot \ln c \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)


ddxex=ex\frac{{d}}{{{d}x}}\;{{e}^x} = {{e}^x}
ddxe()=e()⋅ddx()\frac{{d}}{{{d}x}}\;{{e}^{\left( {\;\;\;\;} \right)}} = {{e}^{\left( {\;\;\;\;} \right)}} \cdot \frac{{d}}{{{d}x}}\left( {\;\;\;\;} \right)
  • Introduction
    ddx2x\frac{d}{{dx}}\;{2^x}
    ddx24x3\frac{d}{{dx}}\;{2^{4{x^3}}}
  • 1.
    ddx35x2\frac{d}{{dx}}\;{3^{{5^{{x^2}}}}}
  • 2.
    ddxex\frac{{d}}{{{d}x}}\;{{e}^x}
    ddxesinx\frac{{d}}{{{d}x}}\;{{e}^{\sin x}}
  • 3.
    Differentiate:
    y=tan(cose5x2)y = {tan\;}(\;\cos {{e}^{5{x^2}}}\;)
  • 4.
    ddxx5\frac{d}{{dx}}\;{x^5} VS. ddx5x\frac{d}{{dx}}\;{5^x}
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25.
Limits and Derivatives
25.1
Finding limits from graphs
25.2
Definition of derivative
25.3
Power rule
25.4
Slope and equation of tangent line
25.5
Chain rule
25.6
Derivative of trigonometric functions
25.7
Derivative of exponential functions
25.8
Product rule
25.9
Quotient rule
25.10
Implicit differentiation
25.11
Derivative of inverse trigonometric functions
25.12
Derivative of logarithmic functions
25.13
Higher order derivatives
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