For the function
$f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1$
Prove that $f^{\prime}(1)=100 f^{\prime}(0)$
The given function is
$f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1$
$\frac{d}{d x} f(x)=\frac{d}{d x}\left[\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1\right]$
$\frac{d}{d x} f(x)=\frac{d}{d x}\left(\frac{x^{100}}{100}\right)+\frac{d}{d x}\left(\frac{x^{99}}{99}\right)+\ldots+\frac{d}{d x}\left(\frac{x^{2}}{2}\right)+\frac{d}{d x}(x)+\frac{d}{d x}(1)$
On using theorem $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$, we obtain
$\frac{d}{d x} f(x)=\frac{100 x^{99}}{100}+\frac{99 x^{98}}{99}+\ldots+\frac{2 x}{2}+1+0$
$=x^{99}+x^{98}+\ldots+x+1$
$\therefore f^{\prime}(x)=x^{99}+x^{98}+\ldots+x+1$
At $x=0$
$f^{\prime}(0)=1$
$f^{\prime}(1)=1^{99}+1^{98}+\ldots+1+1=[1+1+\ldots+1+1]_{100 \mathrm{terms}}=1 \times 100=100$
Thus, $f^{\prime}(1)=100 \times f^{\prime}(0)$
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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,