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Q13:
Find $\frac{dy}{dx}$ of the functions $y^x = x^y$
Find $\frac{dy}{dx}$ of the functions $y^x = x^y$
Solution :
Step 1: Application of Logarithmic Differentiation
Given the implicit function: $y^x = x^y$
To differentiate variables raised to the power of variables, we apply the natural logarithm ($\ln$) to both sides of the equation [Per the property $\ln(a^b) = b \ln a$]:
$\ln(y^x) = \ln(x^y)$
$x \ln y = y \ln x$
Step 2: Differentiating with Respect to $x$
We differentiate both sides with respect to $x$ using the Product Rule, which states $\frac{d}{dx}[u(x)v(x)] = u(x)v'(x) + v(x)u'(x)$ [By the Product Rule of Differentiation]:
$\frac{d}{dx}(x \ln y) = \frac{d}{dx}(y \ln x)$
$x \cdot \frac{d}{dx}(\ln y) + \ln y \cdot \frac{d}{dx}(x) = y \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{d}{dx}(y)$
Step 3: Executing Chain Rule and Implicit Differentiation
Apply the Chain Rule $\frac{d}{dx}[\ln(f(x))] = \frac{1}{f(x)} \cdot f'(x)$ [Per Chain Rule]:
$x \left( \frac{1}{y} \cdot \frac{dy}{dx} \right) + \ln y (1) = y \left( \frac{1}{x} \right) + \ln x \left( \frac{dy}{dx} \right)$
$\frac{x}{y} \frac{dy}{dx} + \ln y = \frac{y}{x} + \ln x \frac{dy}{dx}$
Step 4: Isolating the $\frac{dy}{dx}$ Term
Group all terms containing $\frac{dy}{dx}$ on the left-hand side and constants on the right-hand side:
$\frac{x}{y} \frac{dy}{dx} - \ln x \frac{dy}{dx} = \frac{y}{x} - \ln y$
$\frac{dy}{dx} \left( \frac{x}{y} - \ln x \right) = \frac{y - x \ln y}{x}$
Step 5: Solving for the Final Derivative
Simplify the bracketed term to find the common denominator $y$:
$\frac{dy}{dx} \left( \frac{x - y \ln x}{y} \right) = \frac{y - x \ln y}{x}$
Multiply both sides by $\left( \frac{y}{x - y \ln x} \right)$:
$\frac{dy}{dx} = \frac{y(y - x \ln y)}{x(x - y \ln x)}$
Final Solution: $\frac{dy}{dx} = \frac{y(y - x \ln y)}{x(x - y \ln x)}$
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. $x$. $(x + 3)^2 \cdot (x + 4)^3 \cdot (x + 5)^4$
- Q6: Differentiate the functions w.r.t. $x$. $\left(x + \frac{1}{x}\right)^x + x^{\left(1 + \frac{1}{x}\right)}$
- 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 $x^y + y^x = 1$
- 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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