frac{d}{dx} left( a^{log_{10}(csc^{-1}x)} rig | vedclass

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(B) Let $y = a^{\log_{10}(\csc^{-1}x)}$.Using the derivative formula $\frac{d}{dx}(a^u) = a^u \cdot \ln a \cdot \frac{du}{dx}$,where $u = \log_{10}(\c

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$- a^{\log_{10}(\csc^{-1}x)} \cdot \frac{1}{\csc^{-1}x} \cdot \frac{1}{|x|\sqrt{x^2 - 1}} \cdot \log_{10}a$

Vedclass · Answer

(B) Let $y = a^{\log_{10}(\csc^{-1}x)}$.Using the derivative formula $\frac{d}{dx}(a^u) = a^u \cdot \ln a \cdot \frac{du}{dx}$,where $u = \log_{10}(\csc^{-1}x)$.First,differentiate $u$ with respect to $x$:$\frac{du}{dx} = \frac{d}{dx} \left( \frac{\ln(\csc^{-1}x)}{\ln 10} \right) = \frac{1}{\ln 10} \cdot \frac{1}{\csc^{-1}x} \cdot \frac{d}{dx}(\csc^{-1}x)$.Since $\frac{d}{dx}(\csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2 - 1}}$,we have:$\frac{du}{dx} = \frac{1}{\ln 10} \cdot \frac{1}{\csc^{-1}x} \cdot \left( -\frac{1}{|x|\sqrt{x^2 - 1}} \right)$.Now,substitute back into the derivative of $y$:$\frac{dy}{dx} = a^{\log_{10}(\csc^{-1}x)} \cdot \ln a \cdot \left( \frac{1}{\ln 10} \cdot \frac{1}{\csc^{-1}x} \cdot \left( -\frac{1}{|x|\sqrt{x^2 - 1}} \right) \right)$.Since $\frac{\ln a}{\ln 10} = \log_{10}a$,the expression simplifies to:$\frac{dy}{dx} = - a^{\log_{10}(\csc^{-1}x)} \cdot \frac{1}{\csc^{-1}x} \cdot \frac{1}{|x|\sqrt{x^2 - 1}} \cdot \log_{10}a$.

$\frac{d}{dx} \left( a^{\log_{10}(\csc^{-1}x)} \right) = $

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