Use the rules for derivatives to differentiate the following. y = (log_3 [x (x^2 - ex)] - ln [x...

Question:

Use the rules for derivatives to differentiate the following.

{eq}\displaystyle y = \bigg(\frac{\log_3 [x (x^2 - ex)] - \ln [x \cos x - e^x]}{\tanh^{-1} \sqrt[3] {\frac {\sin x + 1}{x \sin x - 1}}} \bigg)^{\frac {3}{2}} {/eq}.

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Applications of derivatives rules using product, quotient, and power rules.

-The product rule: it is required to find the derivative of the multiplications of two functions as below:

The derivative of the first times the second plus the derivative of the second times the first,

-The quotient rule: it is required to find the derivative of the division of two functions as below:

The derivative of the numerator times the denominator, minus the derivative of the denominator times the numerator, over the denominator squared.

-The power rule: it is required to find the derivative of a function raised to exponent n. If the exponent n is negative, positive or fraction, the power rule is still valid.

Answer and Explanation:

{eq}y' = \frac {3}{2}\bigg(\frac{\log_3 [x (x^2 - ex)] - \ln [x \cos x - e^x]}{\tanh^{-1} \sqrt[3] {\frac {\sin x + 1}{x \sin x - 1}}} \bigg)^{\frac...

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