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Q6:
Differentiate the functions w.r.t. $x$. $\left(x + \frac{1}{x}\right)^x + x^{\left(1 + \frac{1}{x}\right)}$
Differentiate the functions w.r.t. $x$. $\left(x + \frac{1}{x}\right)^x + x^{\left(1 + \frac{1}{x}\right)}$
Solution :
Step 1: Decomposition of the Function
Let the given function be $y = f(x) + g(x)$, where:
- $f(x) = \left(x + \frac{1}{x}\right)^x$
- $g(x) = x^{\left(1 + \frac{1}{x}\right)}$
To find the derivative $\frac{dy}{dx}$, we must compute $\frac{df}{dx}$ and $\frac{dg}{dx}$ separately using logarithmic differentiation [Per the property $\frac{d}{dx}[u^v] = u^v \left( \frac{v}{u} \cdot \frac{du}{dx} + \ln(u) \cdot \frac{dv}{dx} \right)$].
Step 2: Differentiating $f(x) = \left(x + \frac{1}{x}\right)^x$
Taking the natural logarithm on both sides:
$\ln(f) = x \ln\left(x + \frac{1}{x}\right) = x \ln\left(\frac{x^2 + 1}{x}\right) = x [\ln(x^2 + 1) - \ln(x)]$
Differentiating with respect to $x$ using the Product Rule [$\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$]:
$\frac{1}{f} \frac{df}{dx} = (x) \cdot \frac{d}{dx}[\ln(x^2 + 1) - \ln(x)] + [\ln(x^2 + 1) - \ln(x)] \cdot \frac{d}{dx}(x)$
$\frac{1}{f} \frac{df}{dx} = x \left( \frac{2x}{x^2 + 1} - \frac{1}{x} \right) + \ln\left(\frac{x^2 + 1}{x}\right)$
$\frac{df}{dx} = f(x) \left[ \frac{2x^2}{x^2 + 1} - 1 + \ln\left(x + \frac{1}{x}\right) \right] = \left(x + \frac{1}{x}\right)^x \left[ \frac{x^2 - 1}{x^2 + 1} + \ln\left(x + \frac{1}{x}\right) \right]$
Step 3: Differentiating $g(x) = x^{\left(1 + \frac{1}{x}\right)}$
Taking the natural logarithm on both sides:
$\ln(g) = \left(1 + \frac{1}{x}\right) \ln(x)$
Differentiating with respect to $x$:
$\frac{1}{g} \frac{dg}{dx} = \left(1 + \frac{1}{x}\right) \cdot \frac{d}{dx}(\ln x) + \ln(x) \cdot \frac{d}{dx}\left(1 + \frac{1}{x}\right)$
$\frac{1}{g} \frac{dg}{dx} = \left(\frac{x + 1}{x}\right) \cdot \frac{1}{x} + \ln(x) \cdot \left(-\frac{1}{x^2}\right)$
$\frac{dg}{dx} = g(x) \left[ \frac{x + 1}{x^2} - \frac{\ln x}{x^2} \right] = x^{\left(1 + \frac{1}{x}\right)} \left[ \frac{x + 1 - \ln x}{x^2} \right]$
Step 4: Combining Results
Since $y = f(x) + g(x)$, then $\frac{dy}{dx} = \frac{df}{dx} + \frac{dg}{dx}$.
| Term | Result |
|---|---|
| $\frac{df}{dx}$ | $\left(x + \frac{1}{x}\right)^x \left[ \frac{x^2 - 1}{x^2 + 1} + \ln\left(x + \frac{1}{x}\right) \right]$ |
| $\frac{dg}{dx}$ | $x^{\left(1 + \frac{1}{x}\right)} \left[ \frac{x + 1 - \ln x}{x^2} \right]$ |
Final Solution:
$\frac{dy}{dx} = \left(x + \frac{1}{x}\right)^x \left[ \frac{x^2 - 1}{x^2 + 1} + \ln\left(x + \frac{1}{x}\right) \right] + x^{\left(1 + \frac{1}{x}\right)} \left[ \frac{x + 1 - \ln x}{x^2} \right]$
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.5
- Q1: Differentiate the functions w.r.t. $x$. $\cos x \cdot \cos 2x \cdot \cos 3x$
- Q2: Differentiate the functions w.r.t. $x$. $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$
- Q3: Differentiate the functions w.r.t. $x$. $(\log x)^{\cos x}$
- Q4: Differentiate the functions w.r.t. $x$. $x^x - 2^{\sin x}$
- Q5: Differentiate the functions w.r.t. $x$. $(x + 3)^2 \cdot (x + 4)^3 \cdot (x + 5)^4$
- Q7: Differentiate the functions w.r.t. $x$. $(\log x)^x + x^{\log x}$
- Q8: Differentiate the functions w.r.t. $x$. $(\sin x)^x + \sin^{-1} \sqrt{x}$
- Q9: Differentiate the functions w.r.t. $x$. $x^{\sin x} + (\sin x)^{\cos x}$
- Q10: Differentiate the functions w.r.t. $x$. $x^{x \cos x} + \frac{x^2+1}{x^2-1}$
- Q11: Differentiate the functions w.r.t. $x$. $(x \cos x)^x + (x \sin x)^{\frac{1}{x}}$
- Q12: Find $\frac{dy}{dx}$ of the functions $x^y + y^x = 1$
- Q13: Find $\frac{dy}{dx}$ of the functions $y^x = x^y$
- Q14: Find $\frac{dy}{dx}$ of the functions $(\cos x)^y = (\cos y)^x$
- Q15: Find $\frac{dy}{dx}$ of the functions $xy = e^{(x - y)}$
- Q16: Find the derivative of the function given by $f(x) = (1 + x)(1 + x^2)(1 + x^4)(1 + x^8)$ and hence find $f'(1)$.
- Q17(i): Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (i) by using product rule
- Q17(ii): Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (ii) by expanding the product to obtain a single polynomial.
- Q17(iii): Differentiate $(x^2 - 5x + 8)(x^3 + 7x + 9)$ in three ways mentioned below: (iii) by logarithmic differentiation. Do they all give the same answer?
- Q18: If $u, v$ and $w$ are functions of $x$, then show that $\frac{d}{dx} (u \cdot v \cdot w) = \frac{du}{dx} v \cdot w + u \cdot \frac{dv}{dx} \cdot w + u \cdot v \frac{dw}{dx}$ in two ways - first by repeated application of product rule, second by logarithmic differentiation.
CBSE Solutions for Class 12 Mathematics Continuity and Differentiability
- Continuity and Differentiability Class 12 EXERCISE 5.1
- Continuity and Differentiability Class 12 EXERCISE 5.2
- Continuity and Differentiability Class 12 EXERCISE 5.3
- Continuity and Differentiability Class 12 EXERCISE 5.4
- Continuity and Differentiability Class 12 EXERCISE 5.6
- Continuity and Differentiability Class 12 EXERCISE 5.7
Chapters in CBSE - Class 12 Mathematics
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