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Q5:
Find the second order derivatives of the functions. $x^3 \log x$
Find the second order derivatives of the functions. $x^3 \log x$
Solution :
Step 1: Define the Function and First Derivative
Let the given function be denoted by $f(x)$:
$f(x) = x^3 \log x$To find the first derivative, $f'(x)$, we apply the Product Rule of differentiation, which states that for two functions $u(x)$ and $v(x)$, $\frac{d}{dx}[u(x)v(x)] = u'(x)v(x) + u(x)v'(x)$ [Per the Product Rule of Calculus].
Let $u = x^3$ and $v = \log x$. Then:
- $\frac{du}{dx} = 3x^2$
- $\frac{dv}{dx} = \frac{1}{x}$
Applying the rule:
$f'(x) = (3x^2)(\log x) + (x^3)\left(\frac{1}{x}\right)$ $f'(x) = 3x^2 \log x + x^2$Step 2: Determining the Second Order Derivative
The second order derivative, $f''(x)$, is defined as the derivative of the first derivative with respect to $x$:
$f''(x) = \frac{d}{dx} [3x^2 \log x + x^2]$By the Sum Rule of Differentiation, we distribute the derivative across the terms [Per the Linearity of the Derivative Operator]:
$f''(x) = \frac{d}{dx} [3x^2 \log x] + \frac{d}{dx} [x^2]$Step 3: Calculating Individual Components
For the first term, $3x^2 \log x$, we apply the Product Rule again:
$\frac{d}{dx} [3x^2 \log x] = \left(\frac{d}{dx} [3x^2]\right)(\log x) + (3x^2)\left(\frac{d}{dx} [\log x]\right)$ $= (6x)(\log x) + (3x^2)\left(\frac{1}{x}\right)$ $= 6x \log x + 3x$For the second term, $x^2$, we apply the Power Rule, $\frac{d}{dx}[x^n] = nx^{n-1}$ [Per the Power Rule of Differentiation]:
$\frac{d}{dx} [x^2] = 2x$Step 4: Final Algebraic Simplification
Combine the results from Step 3:
$f''(x) = (6x \log x + 3x) + 2x$Combine the like terms ($3x$ and $2x$):
$f''(x) = 6x \log x + 5x$Final Solution: $f''(x) = 6x \log x + 5x$
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 $\log x$
- 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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