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⋅ln⁡c\frac{{d}}{{{d}x}}\;{c^x} = {c^x} \cdot \ln c
ddxc()=c()⋅ln⁡c⋅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}
    ddxesin⁡x\frac{{d}}{{{d}x}}\;{{e}^{\sin x}}
  • 3.
    Differentiate:
    y=tan(cos⁡e5x2)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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26.
Limits and Derivatives
26.1
Finding limits from graphs
26.2
Definition of derivative
26.3
Power rule
26.4
Slope and equation of tangent line
26.5
Chain rule
26.6
Derivative of trigonometric functions
26.7
Derivative of exponential functions
26.8
Product rule
26.9
Quotient rule
26.10
Implicit differentiation
26.11
Derivative of inverse trigonometric functions
26.12
Derivative of logarithmic functions
26.13
Higher order derivatives
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