Derivatives of Exponential Functions

Graph of an Exponential Function

Graph of the exponential function illustrating that its derivative is equal to the value of the function. From any point $P$ on the curve (blue), let a tangent line (red), and a vertical line (green) with height $h$ be drawn, forming a right triangle with a base $b$ on the $x$-axis. Since the slope of the red tangent line (the derivative) at $P$ is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, $h$ must be equal to the ratio of $h$ to $b$. Therefore, the base $b$ must always be $1$.

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Derivatives of Exponential Functions

Graph of an Exponential Function

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