Differentiation Rules

Applying Multiple Differentiation Rules


If f(x)=ln(x+7x2) f(x) = \ln \left(\frac{x + 7}{\sqrt{x-2}}\right) , f(18) f'(18) can be expressed as ab \frac{a}{b} , where aa and bb are coprime positive integers. What is the value of a+ba+b?

The derivative of y=(x9)3(x2+6)2y=(x-9)^3(x^2+6)^2 can be expressed as dydx=(x9)2(x2+6)f(x).\frac{dy}{dx}=(x-9)^2(x^2+6) \cdot f(x). What is f(x)?f(x)?

If y=sin1(x52),\displaystyle y=\sin^{-1}(x^{5}-2), what is dydx?\displaystyle \frac{dy}{dx}?

If the derivative of f(x)=(x+76x+1)5\displaystyle f(x)=\left(\frac{x+7}{6x+1}\right)^{5} is expressed as a(x+7)b(6x+1)6\displaystyle \frac{-a(x+7)^b}{(6x+1)^{6}} for integers aa and bb, what is a+b?a+b?

If y=4xtan1x,\displaystyle y=4x\tan^{-1}\sqrt{x}, what is dydx?\displaystyle \frac{dy}{dx}?


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