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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.

Chain Rule

         

What is the derivative of \(y=4\tan^{4} x?\)

\(f(x)\) is a differentiable function such that \(f(2)=3\) and \(f'(2)=3.\) If \(g(x)=\left(x^2f(x)\right)^2,\) what is the value of \(g'(2)?\)

What is the derivative of \(y=3 ^{x^ 6-4}?\)

Consider the function \(f(x)=\log_{8} \left[ (3x-1)^3 \right] .\) What is the value of \(a\) such that \[f'(a)=\frac{9}{\ln 8}?\]

If the instantaneous rate of change of \(y=\sin^{-1} (3x)\) at \(x=\frac{1}{6}\) can be expressed as \(\frac{18}{\sqrt{a}},\) what is the value of \(a?\)

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