(Solved): URGENT! Please help with question 1 parts a,b,c, and d and question 2 parts a and b Logarithmic diff ...

Logarithmic differentiation is a technique that is useful for differentiating functions that have exponents or which consist of many factors. It involves the derivative of \( \ln \) and elements of implicit differentiation. (a) Let \( y=y(x) \). Use the chain rule to find a formula for \( \frac{d}{d x}(\ln (y(x))) \). The result should involve the functions \( y(x) \) and \( y^{\prime}(x) \). (b) You will compute \( \frac{d}{d x}\left(x^{x}\right) \). Since it is neither a power function, nor an exponential function the result isn't either \( x x^{x-1}=x^{x} \) nor \( \ln (x) x^{x} \). Follow these steps: i. Use the properties of logarithms to express \( \ln \left(x^{x}\right) \) as a product. ii. Set \( y(x)=x^{x} \) and take the logarithm of both sides. Replace the right hand side with the product you found in (1.(b)i.) iii. Differentiate both sides. Replace the left hand side with what you found in (1.a), then solve for \( y^{\prime} \) and substitute \( y(x)=x^{x} \). (c) Let \( f(x)=\frac{x^{7 / 2} \cos (x)}{2^{x}} \). Find \( f^{\prime}(x) \) using logarithmic differentiation. Don't bother simplifying your answer. (Hint: \( \ln (f(x)) \) breaks up into à nice sum of logarithums.) (d) Recall that \( \ln y \) is only defined if \( y \) is positive. How to do logarithmic differentiation if \( y(x) \) is negative? Hint: Check that \( (\ln |x|)^{\prime}=\frac{1}{x} \) even if \( x<0 \). 2. Turns out, the \( \mathrm{CO}_{2} \) molecules in the atmosphere sometimes just fall apart. At pressure of \( P \) atmospheres, a certain fraction \( q \) of a gas decomposes. The quantities \( P \) and \( q \) are related, for some positive constant \( K \), by the equation \[ \frac{4 q^{2} P}{1-q^{2}}=K \] (a) Use implicit differentiation to find \( d q / d P \). (In this case implicit differentiation really is easier than expressing \( q \) through \( P \) explicitly. And you won't get credit for explicit differentiation, anyway.) Hint: You may want to get rid of denominator before yon start differentiating. (b) Show that \( d q / d P<0 \) always. What does this mean in practical terms, i.e. what happens to \( q \) if we increase the pressure? (No essays, please. One sentence is enough.)