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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2.
Differentiation
2.1
Definition of derivative
2.2
Estimating derivatives from a table
2.3
Power rule
2.4
Slope and equation of tangent line
2.5
Chain rule
2.6
Derivative of trigonometric functions
2.7
Derivative of exponential functions
2.8
Product rule
2.9
Quotient rule
2.10
Implicit differentiation
2.11
Derivative of inverse trigonometric functions
2.12
Derivative of logarithmic functions
2.13
Higher order derivatives
Differential Calculus
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