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Differentiation Rules

These are the rules that explain how to take derivatives of any functions: from polynomials to trigonometric functions to logarithms.

Problem Solving - Intermediate

Consider a function \(f(x)\) satisfying \( f(x)=f(4x-2) \) for all real values \(x\). If \(f(x)\) is differentiable for all \(x\) and \(f'(4)=40,\) what is the value of \(f'(54)?\)

Given \[\frac{d}{dx} \ln \frac{x+2}{\sqrt{x-7}}=\frac{f(x)}{2(x+2)(x-7)},\] what is \(f(x)?\)

The polynomial \( P(x) \) satisfies the following identity: \[ P \left( P(x) + x \right) = 11 \left( P(x) + x \right)^2 - 4 \left( P(x) + x \right) + 5 . \] What is the value of \( P'(6)? \)

Given \[g(x)=x \sin^{-1} \left(\frac{x}{24}\right)+\sqrt{576-x^2},\] what is the value of \(g'(12)\)?

Let \(g(x)\) be the inverse function of a differentiable function \(f(x).\) If \(f(3)=7\) and \(f'(3)=\frac{1}{2},\) what is the value of \[\lim_{h \to 0} \frac{g(7+h)-g(7)}{h}?\]

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