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

ddx  cx=cx⋅ln⁡c\frac{{d}}{{{d}x}}\;{c^x} = {c^x} \cdot \ln c
ddx  c(        )=c(        )⋅ln⁡c⋅ddx(        )\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  esin⁡x\frac{{d}}{{{d}x}}\;{{e}^{\sin x}}
  • 3.
    Differentiate:
    y=tan  (  cos⁡e5x2  )y = {tan\;}(\;\cos {{e}^{5{x^2}}}\;)
  • 4.
    ddx  x5\frac{d}{{dx}}\;{x^5} VS. ddx  5x\frac{d}{{dx}}\;{5^x}
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17.
Derivatives
17.1
Definition of derivative
17.2
Power rule
17.3
Gradient and equation of tangent line
17.4
Chain rule
17.5
Derivative of trigonometric functions
17.6
Derivative of exponential functions
17.7
Product rule
17.8
Quotient rule
17.9
Implicit differentiation
17.10
Derivative of inverse trigonometric functions
17.11
Derivative of logarithmic functions
17.12
Higher order derivatives
Higher 2 Maths
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