Derivatives of eˣ and ln x

What the symbols mean

$D$

the value of $f\prime (x)$ at the chosen x

$A$

the coefficient of $e^{kx}$

$k$

the rate in $e^{kx}$

$B$

the coefficient of $\ln x$

$x$

the point (must be positive if there is a ln term)

When to use it

Use it for growth and decay models (the rate of change of $N_0{e}^{kt}$ is k times the amount), for logarithmic functions, and with the chain rule for things like $e^{x^2}$ or $\ln (3x+1)$. The calculator differentiates $A{e}^{kx}+B\ln x$.

Why it works

e is exactly the base whose exponential curve has slope 1 at x = 0, which makes the slope at every point equal to the height. ln x is the inverse of $e^x$, so its slope is the reciprocal: $1/x$.

Common mistakes

• Writing $\frac{d}{dx}{e}^{2x}=e^{2x}$: the chain rule gives an extra factor 2.
• Using the power rule on $e^x$ (giving $x{e}^{x-1}$); the exponent is the variable, not the base.
• Differentiating ln x at x ≤ 0, where ln is not defined.