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  "body": "\n## Statement\n\nThe [[exponential function|0RMVS7RXZX]] is differentiable on $\\mathbb{R}$, and $\\exp'(x) = \\exp(x)$ for all real $x$.\n\n## Remarks\n\nBy [[the sum rule|0E8HNVBCSP]], $\\frac{\\exp(x + h) - \\exp(x)}{h} = \\exp(x) \\cdot \\frac{\\exp(h) - 1}{h}$. The series gives $\\frac{\\exp(h) - 1}{h} \\to 1$ as $h \\to 0$.\nTerm-by-term differentiation of the power series gives the same result.\nThe exponential function is the only solution of $f' = f$ with $f(0) = 1$. Mathlib states the derivative as `Real.hasDerivAt_exp`.\n\n## Explanations\n\n### L1 human\n\nThe slope of the graph of $e^x$ at each point equals its height there.\nNo other function with value $1$ at $0$ has this property.\n",
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