Find the second order derivatives of each of the following functions:
$x^{3} \log x$
$\sqrt{B a s i c}$ Idea: Second order derivative is nothing but derivative of derivative i.e. $\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(\frac{d y}{d x}\right)$
$\sqrt{T h e}$ idea of chain rule of differentiation: If $\mathrm{f}$ is any real-valued function which is the composition of two functions $u$ and $v$, i.e. $f=v(u(x))$. For the sake of simplicity just assume $t=u(x)$
Then $f=v(t) .$ By chain rule, we can write the derivative of $f$ w.r.t to $x$ as:
$\frac{\mathrm{df}}{\mathrm{dx}}=\frac{\mathrm{dv}}{\mathrm{dt}} \times \frac{\mathrm{dt}}{\mathrm{dx}}$
$\sqrt{P r o d u c t}$ rule of differentiation $-\frac{d}{d x}(u v)=u \frac{d v}{d x}+v \frac{d u}{d x}$
Apart from these remember the derivatives of some important functions like exponential, logarithmic, trigonometric etc..
Let's solve now:
Given, $y=x^{3} \log x$
We have to find $\frac{d^{2} y}{d x^{2}}$
As $\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$
So lets first find $d y / d x$ and differentiate it again.
$\therefore \frac{d y}{d x}=\frac{d}{d x}\left(x^{3} \log x\right)$
Let $u=x^{3}$ and $v=\log x$
As, $y=u v$
$\therefore$ Using product rule of differentiation:
$\frac{d y}{d x}=u \frac{d v}{d x}+v \frac{d u}{d x}$
$\therefore \frac{d y}{d x}=x^{3} \frac{d}{d x}(\log x)+\log x \frac{d}{d x} x^{3}$'
$\frac{d y}{d x}=3 x^{2} \log x+\frac{x^{3}}{x}$
$\left[\because \frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})=\frac{1}{\mathrm{x}}\right.$ and $\left.\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}\right]$
Again differentiating w.r.t $x$ :
$\frac{d}{d x}\left(\frac{d y}{d x}\right)=\frac{d}{d x}\left(3 x^{2} \log x+x^{2}\right)$
$=\frac{d}{d x}\left(3 x^{2} \log x\right)+\frac{d}{d x}\left(x^{2}\right)$
Again using the product rule :
$\frac{d^{2} y}{d x^{2}}=3 \log x \frac{d}{d x} x^{2}+3 x^{2} \frac{d}{d x} \log x+\frac{d}{d x} x^{2}$
$\left[\because \frac{d}{d x}(\log x)=\frac{1}{x}\right.$ and $\left.\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}\right]$
$\frac{d^{2} y}{d x^{2}}=6 x \log x+\frac{3 x^{2}}{x}+2 x$
$\frac{d^{2} y}{d x^{2}}=6 x \log x+5 x$
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All Study Material
- JEE Main
- Exam Pattern
- Previous Year Papers
- PYQ Chapterwise
- Physics
- Kinematics 1D
- Kinemetics 2D
- Friction
- Work, Power, Energy
- Centre of Mass and Collision
- Rotational Dynamics
- Gravitation
- Calorimetry
- Elasticity
- Thermal Expansion
- Heat Transfer
- Kinetic Theory of Gases
- Thermodynamics
- Simple Harmonic Motion
- Wave on String
- Sound waves
- Fluid Mechanics
- Electrostatics
- Current Electricity
- Capacitor
- Magnetism and Matter
- Electromagnetic Induction
- Atomic Structure
- Dual Nature of Matter
- Nuclear Physics
- Radioactivity
- Semiconductors
- Communication System
- Error in Measurement & instruments
- Alternating Current
- Electromagnetic Waves
- Wave Optics
- X-Rays
- All Subjects
- Physics
- Motion in a Plane
- Law of Motion
- Work, Energy and Power
- Systems of Particles and Rotational Motion
- Gravitation
- Mechanical Properties of Solids
- Mechanical Properties of Fluids
- Thermal Properties of matter
- Thermodynamics
- Kinetic Theory
- Oscillations
- Waves
- Electric Charge and Fields
- Electrostatic Potential and Capacitance
- Current Electricity
- Thermoelectric Effects of Electric Current
- Heating Effects of Electric Current
- Moving Charges and Magnetism
- Magnetism and Matter
- Electromagnetic Induction
- Alternating Current
- Electromagnetic Wave
- Ray Optics and Optical Instruments
- Wave Optics
- Dual Nature of Radiation and Matter
- Atoms
- Nuclei
- Semiconductor Electronics: Materials, Devices and Simple Circuits.
- Chemical Effects of Electric Current,