Question

If \[ y= \left( x^{\sin x} \right)^{\tan x}, \] then find \[ \dfrac{dy}{dx}. \]

• A. [y[ \sec^2 x \sin x \ln x + \tan x \cos x \ln x + \dfrac{\tan x \sin x}{x}]\]
• B. [y(\cos x + \sin x)]
• C. [y \tan x]
• D. [ x^{\sin x}\]

Hint:

For nested exponential expressions involving trigonometric exponents, logarithmic differentiation is almost always the fastest and safest method.

Answer 1

The Correct Option is A

Concept: This is a highly composite logarithmic differentiation problem where the variable appears:
• in the base,
• inside trigonometric functions,
• and also in the exponent. Such expressions are simplified only through repeated logarithmic transformation and systematic implicit differentiation.

Step 1: Simplifying the given function.
Given: \[ y= \left( x^{\sin x} \right)^{\tan x} \] Using: \[ (a^m)^n=a^{mn} \] we get: \[ y = x^{\sin x\tan x} \] Now observe: \[ \sin x\tan x = \sin x\left(\frac{\sin x}{\cos x}\right) = \frac{\sin^2x}{\cos x} \] However, keeping the exponent in product form is more convenient for differentiation. Thus, \[ y=x^{\sin x\tan x} \]

Step 2: Taking logarithm on both sides.
Applying natural logarithm: \[ \ln y = \sin x\tan x\ln x \] Now the expression becomes a product of three functions: \[ (\sin x)(\tan x)(\ln x) \]

Step 3: Differentiating implicitly.
Differentiate both sides with respect to \(x\): \[ \dfrac{1}{y}\dfrac{dy}{dx} = \dfrac{d}{dx} [ \sin x\tan x\ln x ] \] Apply product rule carefully for three factors.

Step 4: Differentiating each factor.
Using: \[ \dfrac{d}{dx}(\sin x)=\cos x \] \[ \dfrac{d}{dx}(\tan x)=\sec^2x \] \[ \dfrac{d}{dx}(\ln x)=\frac1x \] Hence, \[ \dfrac{1}{y}\dfrac{dy}{dx} = (\cos x)(\tan x)(\ln x) + (\sin x)(\sec^2x)(\ln x) + (\sin x)(\tan x)\left(\frac1x\right) \] Rearranging terms: \[ \dfrac{1}{y}\dfrac{dy}{dx} = \sec^2x\sin x\ln x + \tan x\cos x\ln x + \dfrac{\tan x\sin x}{x} \]

Step 5: Obtaining the final derivative.
Multiply both sides by \(y\): \[ \dfrac{dy}{dx} = y \left[ \sec^2x\sin x\ln x + \tan x\cos x\ln x + \dfrac{\tan x\sin x}{x} \right] \] Therefore, \[ \boxed{ \dfrac{dy}{dx} = y \left[ \sec^2x\sin x\ln x + \tan x\cos x\ln x + \dfrac{\tan x\sin x}{x} \right] } \]

Step 6: Why ordinary differentiation becomes difficult.
If logarithmic differentiation were not used, direct differentiation would involve repeated exponential chain rules and implicit structures simultaneously. Logarithmic differentiation converts complicated exponential expressions into manageable algebraic differentiation.