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If f(x)=ln(x+7x−2) f(x) = \ln \left(\frac{x + 7}{\sqrt{x-2}}\right) f(x)=ln(x−2x+7), f′(18) f'(18) f′(18) can be expressed as ab \frac{a}{b} ba, where aaa and bbb are coprime positive integers. What is the value of a+ba+ba+b?
The derivative of y=(x−9)3(x2+6)2y=(x-9)^3(x^2+6)^2y=(x−9)3(x2+6)2 can be expressed as dydx=(x−9)2(x2+6)⋅f(x).\frac{dy}{dx}=(x-9)^2(x^2+6) \cdot f(x).dxdy=(x−9)2(x2+6)⋅f(x). What is f(x)?f(x)?f(x)?
If y=sin−1(x5−2),\displaystyle y=\sin^{-1}(x^{5}-2),y=sin−1(x5−2), what is dydx?\displaystyle \frac{dy}{dx}?dxdy?
If the derivative of f(x)=(x+76x+1)5\displaystyle f(x)=\left(\frac{x+7}{6x+1}\right)^{5}f(x)=(6x+1x+7)5 is expressed as −a(x+7)b(6x+1)6\displaystyle \frac{-a(x+7)^b}{(6x+1)^{6}}(6x+1)6−a(x+7)b for integers aaa and bbb, what is a+b?a+b?a+b?
If y=4xtan−1x,\displaystyle y=4x\tan^{-1}\sqrt{x},y=4xtan−1x, what is dydx?\displaystyle \frac{dy}{dx}?dxdy?
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