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Q12:
Find $\frac{dy}{dx}$ of the functions $x^y + y^x = 1$
Find $\frac{dy}{dx}$ of the functions $x^y + y^x = 1$
Solution :
Step 1: Decomposition of the Implicit Equation
Given the implicit equation: $x^y + y^x = 1$.
To differentiate this with respect to $x$, we must define two separate functions, $u = x^y$ and $v = y^x$, such that:
$u + v = 1$
Differentiating with respect to $x$ yields:
$\frac{du}{dx} + \frac{dv}{dx} = 0$ [Per the Sum Rule of Differentiation] --- (Equation 1)
Step 2: Logarithmic Differentiation of $u = x^y$
Taking the natural logarithm on both sides:
$\ln(u) = \ln(x^y) = y \ln(x)$ [By Logarithmic Power Rule]
Differentiating with respect to $x$ using the product rule:
$\frac{1}{u} \frac{du}{dx} = y \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{dy}{dx}$
$\frac{1}{u} \frac{du}{dx} = \frac{y}{x} + \ln x \frac{dy}{dx}$
$\frac{du}{dx} = x^y \left( \frac{y}{x} + \ln x \frac{dy}{dx} \right)$ --- (Equation 2)
Step 3: Logarithmic Differentiation of $v = y^x$
Taking the natural logarithm on both sides:
$\ln(v) = \ln(y^x) = x \ln(y)$
Differentiating with respect to $x$ using the product rule:
$\frac{1}{v} \frac{dv}{dx} = x \cdot \frac{d}{dx}(\ln y) + \ln y \cdot \frac{d}{dx}(x)$
$\frac{1}{v} \frac{dv}{dx} = x \cdot \frac{1}{y} \frac{dy}{dx} + \ln y \cdot (1)$
$\frac{dv}{dx} = y^x \left( \frac{x}{y} \frac{dy}{dx} + \ln y \right)$ --- (Equation 3)
Step 4: Algebraic Substitution and Solving for $\frac{dy}{dx}$
Substitute Equation 2 and Equation 3 into Equation 1:
$x^y \left( \frac{y}{x} + \ln x \frac{dy}{dx} \right) + y^x \left( \frac{x}{y} \frac{dy}{dx} + \ln y \right) = 0$
Expand the terms:
$x^y \cdot \frac{y}{x} + x^y \ln x \frac{dy}{dx} + y^x \frac{x}{y} \frac{dy}{dx} + y^x \ln y = 0$
Factor out $\frac{dy}{dx}$:
$\frac{dy}{dx} (x^y \ln x + x y^{x-1}) = -(y x^{y-1} + y^x \ln y)$
Isolate $\frac{dy}{dx}$:
$\frac{dy}{dx} = -\frac{y x^{y-1} + y^x \ln y}{x^y \ln x + x y^{x-1}}$
Summary Illustration
Final Solution: \frac{dy}{dx} = -\frac{y x^{y-1} + y^x \ln y}{x^y \ln x + x y^{x-1}}
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. $x$. $(x \cos x)^x + (x \sin x)^{\frac{1}{x}}$
- 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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