av David Kedrowski för 14 årar sedan
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Constant Rule
d/dx[e^e] = 0
Power Rule
d/dx[x^e] = e x^{e-1}
Exponential Rule
d/dx[e^x] = e^x
Logarithmic differentiation
d/dx[x^x] = x^x (1 + ln x)
When confronted with an integral of the form
S a^x dx
there are two choices.
One is to convert the exponential expression with base a to an equivalent exponential expression with base e. That is, consider
S e^{(ln a)x} dx
remembering that ln a is a constant.
The second option is to use the following integration formula,
S a^x dx = (1/ln a) a^x + C
Let a be a positive real number (a<>1) and let u be a differentiable function of x.
d
dx
d du
dx dx
d 1
dx (ln a) x
d 1 du
dx (ln a) u dx
The logarithm with base 10.
If a is a positive real number (a<>1) and x is any positive real number, then the logarithmic function to the base a is denoted by log_a x and is defined as
log_a x = (1/ln a) ln x
If a is a positive real number (a<>1) and x is any real number, then the exponential function to the base a is denoted by a^x and is defined by
a^x = e^{(ln a)x}
If a=1, then y=1^x=1 is a constant function.
