If f^''(x)=(cos(logx))/(x), f^'(1)=0 and y=f((2 x+3)/(3-2 x)) then (d y)/(d x) is equal to (a) (

step 1 we have problem number one hundred ninety six given that if f x equal to sine log x f x equal to

If f^''(x)=(cos(logx))/(x), f^'(1)=0 and y=f((2 x+3)/(3-2...

If $f^{\prime \prime}(x)=\frac{\cos (\log x)}{x}, f^{\prime}(1)=0$ and $y=f\left(\frac{2 x+3}{3-2 x}\right)$ then $\frac{d y}{d x}$ is equal to (a) $(\sin (\log x)) \frac{1}{\cos x}$ (b) $\sin \left(\log \left(\frac{2 x+3}{3-2 x}\right)\right)$ (c) $\frac{12}{(3-2 x)^{2}} \sin \left(\log \left(\frac{2 x+3}{3-2 x}\right)\right)$ (d) None of these

If $f^{\prime \prime}(x)=\frac{\cos (\log x)}{x}, f^{\prime}(1)=0$ and $y=f\left(\frac{2 x+3}{3-2 x}\right)$
then $\frac{d y}{d x}$ is equal to
(a) $(\sin (\log x)) \frac{1}{\cos x}$
(b) $\sin \left(\log \left(\frac{2 x+3}{3-2 x}\right)\right)$
(c) $\frac{12}{(3-2 x)^{2}} \sin \left(\log \left(\frac{2 x+3}{3-2 x}\right)\right)$
(d) None of these

Key Concepts

Integration of Higher Order Derivatives

When given a higher order derivative with an initial condition, as in the case of a second derivative with a given first derivative value, one must integrate to recover the lower order derivative or the original function. This process involves techniques such as substitution when the derivative contains composite functions like logarithms or trigonometric functions of logarithms.

Differentiation of Rational Functions

Rational functions, which are quotients of two functions, often require the use of the quotient rule for their differentiation. This method is used to find the derivative of a function defined as a ratio, ensuring that both the numerator and denominator are appropriately accounted for in the derivative.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. It states that if you have a function composed of an outer function and an inner function, then the derivative of the composite function is the derivative of the outer function (evaluated at the inner function) multiplied by the derivative of the inner function.

Composite Functions

Composite functions are created when one function is applied to the result of another function. Understanding these functions is crucial because their differentiation involves applying the chain rule, as the rate of change of the outer function depends on the inner function.

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