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Q14:
Find $\frac{dy}{dx}$ of the functions $(\cos x)^y = (\cos y)^x$
Find $\frac{dy}{dx}$ of the functions $(\cos x)^y = (\cos y)^x$
Solution :
Step 1: Taking the Logarithm of Both Sides
Given the equation: $(\cos x)^y = (\cos y)^x$
To differentiate functions where variables appear in both the base and the exponent, we apply the natural logarithm ($\ln$) to both sides [By the property of logarithmic functions]:
$\ln((\cos x)^y) = \ln((\cos y)^x)$
Using the logarithmic power rule $\ln(a^b) = b \ln(a)$, we simplify the expression:
$y \ln(\cos x) = x \ln(\cos y)$
Step 2: Implicit Differentiation
We differentiate both sides with respect to $x$, applying the product rule $\frac{d}{dx}[uv] = u \frac{dv}{dx} + v \frac{du}{dx}$ and the chain rule [Per the Chain Rule for differentiation]:
$\frac{d}{dx}[y \ln(\cos x)] = \frac{d}{dx}[x \ln(\cos y)]$
$y \cdot \frac{d}{dx}[\ln(\cos x)] + \ln(\cos x) \cdot \frac{dy}{dx} = x \cdot \frac{d}{dx}[\ln(\cos y)] + \ln(\cos y) \cdot \frac{d}{dx}[x]$
Step 3: Evaluating Derivatives
Applying the derivative of $\ln(u)$ which is $\frac{1}{u} \cdot \frac{du}{dx}$:
$y \cdot \left( \frac{1}{\cos x} \cdot (-\sin x) \right) + \ln(\cos x) \frac{dy}{dx} = x \cdot \left( \frac{1}{\cos y} \cdot (-\sin y) \frac{dy}{dx} \right) + \ln(\cos y) \cdot 1$
Simplifying the trigonometric terms ($\frac{\sin \theta}{\cos \theta} = \tan \theta$):
$-y \tan x + \ln(\cos x) \frac{dy}{dx} = -x \tan y \frac{dy}{dx} + \ln(\cos y)$
Step 4: Isolating the $\frac{dy}{dx}$ Terms
Group all terms containing $\frac{dy}{dx}$ on the left-hand side and the remaining terms on the right-hand side:
$\ln(\cos x) \frac{dy}{dx} + x \tan y \frac{dy}{dx} = \ln(\cos y) + y \tan x$
Factor out $\frac{dy}{dx}$:
$\frac{dy}{dx} [\ln(\cos x) + x \tan y] = \ln(\cos y) + y \tan x$
Step 5: Final Algebraic Solution
Divide both sides by the bracketed coefficient to solve for $\frac{dy}{dx}$:
$\frac{dy}{dx} = \frac{\ln(\cos y) + y \tan x}{\ln(\cos x) + x \tan y}$
Final Solution: $\frac{dy}{dx} = \frac{\ln(\cos y) + y \tan x}{\ln(\cos x) + x \tan y}$
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 $x^y + y^x = 1$
- Q13: Find $\frac{dy}{dx}$ of the functions $y^x = x^y$
- 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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