Exponential derivatives

01A rate proportional to the value#

An exponential changes at a rate proportional to its current value. This is why exponentials appear in growth models and in the scores that become ML probabilities. The constant base determines the proportionality factor.

For the natural exponential $f(x)=e^x$, the derivative equals the function: $f'(x)=e^x$. At $x=0$, its value and slope are both $1$. At a larger input, both are larger.

A different positive base has a different rate. Write $a^x=e^{x\ln a}$ and use the chain rule. The inner derivative contributes $\ln a$:

02Include the inner change#

For $f(x)=2^{3x}$, the outer exponential contributes $2^{3x}\ln2$, and the inner function $3x$ contributes $3$.

At $x=0$, the derivative is $3\ln2$, not $1$.

Likewise, $e^{-2x}$ has derivative $-2e^{-2x}$. Its values stay positive but decrease as $x$ increases. The minus sign describes the direction of change, not the sign of the function.

03Check the base#

For a constant base $a>0$, the real exponential rule applies for all real inputs. When $a=1$, the function is constant and the derivative is zero because $\ln1=0$. For $0<a<1$, $\ln a<0$ and the function decreases.

Do not use the power rule as if the exponent were a fixed number. In $x^3$, the base varies. In $3^x$, the exponent varies. That distinction determines which derivative rule you need.