Additional Math – Differentiation – Exponential (e) and Trigo (Cosine)

$latex \displaystyle y=\ \ln \left( {\frac{{{{e}^{{{{x}^{2}}}}}}}{{\cos (1-x)}}} \right)$

Before you Differentiate, simplify the expression

using the Laws of Logarithm   $latex \displaystyle {{\log }_{a}}(\frac{x}{y})\ =\log {}_{a}x-{{\log }_{a}}y$

$latex \displaystyle y=\text{ln}\left( {{{e}^{{{{x}^{2}}}}}} \right)-\ln \left[ {\cos (1-x)\ } \right]$

and Laws of Logarithm again  $latex \displaystyle {{\log }_{a}}{{x}^{c}}\ =c\log {}_{a}x$

$latex \displaystyle y={{x}^{2}}\text{ln}\left( e \right)-\ln \left[ {\cos (1-x)\ } \right]$

$latex \displaystyle y={{x}^{2}}-\ln \left[ {\cos (1-x)\ } \right]$

Differentiate the expression and simplify.

$latex \displaystyle \frac{{dy}}{{dx}}=\frac{{{{x}^{3}}}}{3}-\frac{1}{{\cos (1-x)\ }}\times -\sin (1-x)\times -1$

$latex \displaystyle \frac{{dy}}{{dx}}=\frac{{{{x}^{3}}}}{3}-\frac{{\sin (1-x)}}{{\cos (1-x)\ }}$

$latex \displaystyle \frac{{dy}}{{dx}}=\frac{{{{x}^{3}}}}{3}-\tan (1-x)$


You can differentiate without first simplifying, but you will find that you will take more steps to get the answer. REMEMBER: The More (unnecessary) Steps = The More Mistakes.  See Post about Differentiation of ln with sine


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