Definition
Definition
The e-function, also called the exponential function, has the form f(x) = e^x. Here, e is a mathematical constant, approximately 2.71828.
The special thing about the e-function is that its derivative is identical to the function itself. This means:
f'(x) = e^x
Scheme for the derivation of exponential functions
Scheme
To differentiate an exponential function of the form f(x) = a \cdot e^{b \cdot x} multiply by the constant b. The differentiation rule is:
f'(x) = a \cdot b \cdot e^{b \cdot x}
Besides the e-function, there are also other exponential functions, such as f(x) = a^x. The derivative of such functions is:
f'(x) = a^x \cdot \ln(a)
Example for the derivation of exponential functions
Function: f(x) = 2^x
Derivative: f'(x) = 2^x \cdot \ln(2)
Function: g(x) = 3 \cdot e^{2x}
Derivative: g'(x) = 6 \cdot e^{2x}
Summary
Summary
- ★The derivative of the e-function is e^x.
- ★For other exponential functions a^x, the natural logarithm of the base, i.e. \ln(a), is multiplied.
Exercises
1 / 2
Differentiate
f(x) = 5 \cdot e^{-3x} .
2 / 2
Differentiate
g(x) = -2 \cdot 1{,}5^{-x} .
rockettutor.com
Start now & improve your maths grades! 🚀
Cancel your trial online at any time