Calculus
Differentiation Rules
If \( f(x) = \ln \left(\frac{x + 7}{\sqrt{x-2}}\right) \), \( f'(18) \) can be expressed as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime positive integers. What is the value of \(a+b\)?
The derivative of \(y=(x-9)^3(x^2+6)^2\) can be expressed as \(\frac{dy}{dx}=(x-9)^2(x^2+6) \cdot f(x).\) What is \(f(x)?\)
If \(\displaystyle y=\sin^{-1}(x^{5}-2),\) what is \(\displaystyle \frac{dy}{dx}?\)
If the derivative of \(\displaystyle f(x)=\left(\frac{x+7}{6x+1}\right)^{5}\) is expressed as \(\displaystyle \frac{-a(x+7)^b}{(6x+1)^{6}}\) for integers \(a\) and \(b\), what is \(a+b?\)
If \(\displaystyle y=4x\tan^{-1}\sqrt{x},\) what is \(\displaystyle \frac{dy}{dx}?\)