Cauchy's Mean Value Theorem

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in final step, is the tangent line slope calculation mean to say f(x)dx /g(x) dx evaluated at c (not the derivatives) thank you

HannahDavies-dpqn
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But this proof does not specify that c is inbetween a and b right? I found another one with rolled theorem that accounts for that unless im mistaken.

ZEPHYRZHANG-mgzi