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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.

Quotient Rule

         

If \( f(x) = \frac{2x + 4}{3x + 7} \), what is the value of \(f'(3)\)?

If \(f(x)=\frac{-9x}{x^2+5} \), what is the value of \(f'(5)\)?

Let \(g(x)\) be a differentiable function such that \(g(0)=2.\) If \(\displaystyle f(x)=\frac{1}{xg(x)+6},\) where what is the value of \(f'(0)?\)

Let \(g(x)\) be a differentiable function such that \(g(x) \neq -7.\) If \(f(x)=\frac{x}{7+g(x)}\) and \(f'(0)=\frac{1}{14},\) what is the value of \(g(0)?\)

If \(a\) and \(b\) are constants such that \(f(x)=\frac{ax+b}{x^2+x+1}\) and \[f'(0)=-3, \; f'(-1)=10,\] what is \(2a+b?\)

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