If y = sin (xˣ) + x^(tan x), find dy/dx.

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Q. If y = sin (xˣ) + x^(tan x), find dy/dx.

Goa GbshseGBSHSE Class 12 Board Exam 2018Subjective· 4mImportance★★★★★

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Differentiate each term separately using the chain rule and logarithmic differentiation for the exponential-variable terms.

$y = \sin(x^x) + x^{\tan x}$

Term 1: Let $u=x^x$ . Taking logs: $\log u = x\log x \Rightarrow \dfrac1u\dfrac{du}{dx} = \log x + 1$

$\Rightarrow \dfrac{du}{dx} = x^x(1+\log x)$

So $\dfrac{d}{dx}\sin(x^x) = \cos(x^x)\cdot x^x(1+\log x)$ (chain rule)

Term 2: Let $v=x^{\tan x}$ . Taking logs: $\log v = \tan x\log x \Rightarrow \dfrac1v\dfrac{dv}{dx} = \sec^2x\log x + \dfrac{\tan x}{x}$

…

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2026Set A1 markMCQ
$y = x^x$
(a) $x^x(\log x + 1)$
(b) $\log x$
(c) $(\log x + 1)$
(d) $nx^{n-1}$
›Reveal solutionSolution

$\frac{dy}{dx} = x^x(\log x + 1)$ .

Take logs of $y = x^x$ :

$\log y = x\log x.$

Differentiate implicitly:

$\frac{1}{y}\frac{dy}{dx} = \log x + x\cdot\frac{1}{x} = \log x + 1.$

So …

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CBSE 2026Set SEM31 markMCQ
$x^m y^n = (x + y)^{m+n}$
(a) $0$
(b) $\dfrac{y}{x}$
(c) $\dfrac{x + y}{xy}$
(d) $xy$
›Reveal solutionSolution

Use logarithmic differentiation on $x^m y^n=(x+y)^{m+n}$ ; the cross terms cancel and $\dfrac{dy}{dx}=\dfrac{y}{x}$ .

Logarithmic differentiation of an implicit relation is a standard NCERT/CBSE Class 12 continuity and differentiability method.

Take natural logs of both sides:

$m\ln x + n\ln y = (m+n)\ln(x+y).$

Differentiate with respect to $x$ :

$\frac{m}{x} + \frac{n}{y}\frac{dy}{dx} = (m+n)\cdot\frac{1+\tfrac{dy}{dx}}{x+y}.$

Multiply out and collect. Writing $y'=\tfrac{dy}{dx}$ : …

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CBSE 2025Set ANNUAL1 markMCQ
$Q. The derivative of x^(2x) with respect to x is ......................$
(a) x^(2x−1)
(b) 2x^(2x) log x
(c) 2x^(2x)(1 + log x)
(d) 2x^(2x)(1 − 2 log x)
›Reveal solutionSolution

When the variable appears in both the base and the exponent, take logarithms first (logarithmic differentiation), then differentiate implicitly.

Given: $y = x^{2x}$

Step 1 — take log of both sides:

$\ln y = 2x \ln x$

Step 2 — differentiate both sides w.r.t. $x$ (product rule on the right):

$\frac{1}{y}\frac{dy}{dx} = 2\ln x + 2x \cdot \frac{1}{x} = 2\ln x + 2$

…

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←Previous chapterCh 4Determinants

Q.If y = sin (xˣ) + x^(tan x), find dy/dx.

Concept understanding — Logarithmic Differentiation

Q.If y = sin (xˣ) + x^(tan x), find dy/dx.

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