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Q8:
Find the second order derivatives of the functions. $\tan^{-1} x$
Find the second order derivatives of the functions. $\tan^{-1} x$
Solution :
Step 1: Identifying the Function and First Order Derivative
Given the function $f(x) = \tan^{-1} x$. To find the second order derivative, we must first determine the first order derivative with respect to $x$.
Applying the standard derivative formula for inverse trigonometric functions [By the Rule of Derivatives of Inverse Functions]:
$$\frac{d}{dx} (\tan^{-1} x) = \frac{1}{1 + x^2}$$
Let $y = \tan^{-1} x$. Therefore, the first derivative is:
$$y' = \frac{dy}{dx} = (1 + x^2)^{-1}$$
Step 2: Applying the Chain Rule for Second Order Differentiation
To find the second order derivative $\frac{d^2y}{dx^2}$, we differentiate $y'$ with respect to $x$ [By the Chain Rule of Differentiation]:
$$\frac{d^2y}{dx^2} = \frac{d}{dx} \left[ (1 + x^2)^{-1} \right]$
Using the Power Rule $\frac{d}{dx}(u^n) = n u^{n-1} \cdot \frac{du}{dx}$, where $u = 1 + x^2$ and $n = -1$:
$$\frac{d^2y}{dx^2} = -1 \cdot (1 + x^2)^{-1 - 1} \cdot \frac{d}{dx}(1 + x^2)$$
Step 3: Evaluating the Differential Component
Calculating the derivative of the inner function $(1 + x^2)$ with respect to $x$ [By the Sum Rule and Power Rule of Differentiation]:
$$\frac{d}{dx}(1 + x^2) = 0 + 2x = 2x$$
Substituting this result back into our expression for the second derivative:
$$\frac{d^2y}{dx^2} = -1 \cdot (1 + x^2)^{-2} \cdot (2x)$$
Step 4: Simplifying the Final Expression
Combining the terms into a standard rational algebraic form:
$$\frac{d^2y}{dx^2} = \frac{-2x}{(1 + x^2)^2}$$
Final Solution: The second order derivative of $\tan^{-1} x$ is $\frac{-2x}{(1 + x^2)^2}$.
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. $e^{6x} \cos 3x$
- 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 $\frac{d^2y}{dx^2} + y = 0$
- Q12: If $y = \cos^{-1} x$, Find $\frac{d^2y}{dx^2}$ in terms of $y$ alone.
- 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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