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Q12:
If $y = \cos^{-1} x$, Find $\frac{d^2y}{dx^2}$ in terms of $y$ alone.
Solution :
Step 1: First Order Differentiation
Given the function: $y = \cos^{-1} x$
To find the first derivative, differentiate both sides with respect to $x$ using the standard derivative formula for inverse trigonometric functions [Derivative of $\cos^{-1}x = -\frac{1}{\sqrt{1-x^2}}$]:
$\frac{dy}{dx} = -\frac{1}{\sqrt{1-x^2}}$
We can rewrite this as: $\frac{dy}{dx} = -(1-x^2)^{-1/2}$
Step 2: Second Order Differentiation
Differentiate $\frac{dy}{dx}$ with respect to $x$ again to obtain $\frac{d^2y}{dx^2}$. Apply the Power Rule $\frac{d}{dx}[u^n] = nu^{n-1} \cdot \frac{du}{dx}$ and the Chain Rule:
$\frac{d^2y}{dx^2} = \frac{d}{dx} [-(1-x^2)^{-1/2}]$
$\frac{d^2y}{dx^2} = - \left( -\frac{1}{2} \right) (1-x^2)^{-3/2} \cdot \frac{d}{dx}(1-x^2)$ [By Chain Rule]
$\frac{d^2y}{dx^2} = \frac{1}{2} (1-x^2)^{-3/2} \cdot (-2x)$
$\frac{d^2y}{dx^2} = -x(1-x^2)^{-3/2} = -\frac{x}{(1-x^2)^{3/2}}$
Step 3: Conversion to Terms of $y$
From the original equation $y = \cos^{-1} x$, we know that $x = \cos y$.
Substitute $x = \cos y$ into the expression for the second derivative:
$\frac{d^2y}{dx^2} = -\frac{\cos y}{(1-\cos^2 y)^{3/2}}$
Applying the Pythagorean identity $[\sin^2 y + \cos^2 y = 1 \implies 1-\cos^2 y = \sin^2 y]$:
$\frac{d^2y}{dx^2} = -\frac{\cos y}{(\sin^2 y)^{3/2}}$
$\frac{d^2y}{dx^2} = -\frac{\cos y}{(\sin y)^3}$
Step 4: Algebraic Simplification
Decompose the denominator into $\sin y \cdot \sin^2 y$:
$\frac{d^2y}{dx^2} = -\left( \frac{\cos y}{\sin y} \right) \cdot \left( \frac{1}{\sin^2 y} \right)$
Using trigonometric identities $[\cot y = \frac{\cos y}{\sin y}$ and $\csc^2 y = \frac{1}{\sin^2 y}]$:
$\frac{d^2y}{dx^2} = -\cot y \csc^2 y$
Final Solution: \frac{d^2y}{dx^2} = -\cot y \csc^2 y
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.7
- Q1: Find the second order derivatives of the functions $x^2 + 3x + 2$
- Q2: Find the second order derivatives of the functions $x^{20}$
- Q3: Find the second order derivatives of the functions $x \cdot \cos x$
- Q4: Find the second order derivatives of the functions $\log x$
- Q5: Find the second order derivatives of the functions. $x^3 \log x$
- Q6: Find the second order derivatives of the functions. $e^x \sin 5x$
- Q7: Find the second order derivatives of the functions. $e^{6x} \cos 3x$
- Q8: Find the second order derivatives of the functions. $\tan^{-1} x$
- Q9: Find the second order derivatives of the functions. $\log(\log x)$
- Q10: Find the second order derivatives of the functions. $\sin(\log x)$
- Q11: If $y = 5 \cos x - 3 \sin x$, prove that $\frac{d^2y}{dx^2} + y = 0$
- Q13: If $y = 3 \cos(\log x) + 4 \sin(\log x)$, show that $x^2 y_2 + xy_1 + y = 0$
- Q14: If $y = Ae^{mx} + Be^{nx}$, show that $\frac{d^2y}{dx^2} - (m + n)\frac{dy}{dx} + mny = 0$
- Q15: If $y = 500e^{7x} + 600e^{-7x}$, show that $\frac{d^2y}{dx^2} = 49y$
- Q16: If $e^y (x + 1) = 1$, show that $\frac{d^2y}{dx^2} = \left(\frac{dy}{dx}\right)^2$
- Q17: If $y = (\tan^{-1} x)^2$, show that $(x^2 + 1)^2 y_2 + 2x(x^2 + 1)y_1 = 2$
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 5.5
- Continuity and Differentiability Class 12 EXERCISE 5.6
Chapters in CBSE - Class 12 Mathematics
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