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Q2:
Find the second order derivatives of the functions $x^{20}$
Find the second order derivatives of the functions $x^{20}$
Solution :
Step 1: Define the Function
Let the given function be denoted by $y$:
$y = f(x) = x^{20}$
Step 2: First Order Differentiation
To find the first derivative $\frac{dy}{dx}$, we apply the Power Rule of differentiation, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$ [By Power Rule of Calculus].
$\frac{dy}{dx} = \frac{d}{dx}(x^{20})$
$\frac{dy}{dx} = 20x^{20-1}$
$\frac{dy}{dx} = 20x^{19}$
Step 3: Second Order Differentiation
The second order derivative, denoted as $\frac{d^2y}{dx^2}$, is obtained by differentiating the first derivative with respect to $x$:
$\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)$
$\frac{d^2y}{dx^2} = \frac{d}{dx}(20x^{19})$
Applying the Constant Multiple Rule [$\frac{d}{dx}(cf(x)) = c \frac{d}{dx}f(x)$]:
$\frac{d^2y}{dx^2} = 20 \cdot \frac{d}{dx}(x^{19})$
Applying the Power Rule again:
$\frac{d^2y}{dx^2} = 20 \cdot (19x^{19-1})$
$\frac{d^2y}{dx^2} = 20 \cdot 19 \cdot x^{18}$
Step 4: Algebraic Simplification
Perform the scalar multiplication of the constants:
$20 \times 19 = 380$
Therefore:
$\frac{d^2y}{dx^2} = 380x^{18}$
Visual Representation of Power Function Growth
Final Solution: The second order derivative of $x^{20}$ is $380x^{18}$.
More Questions from Class 12 Mathematics Continuity and Differentiability EXERCISE 5.7
- Q1: Find the second order derivatives of the functions $x^2 + 3x + 2$
- 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 $\frac{d^2y}{dx^2} + y = 0$
- 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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