Find the derivation of each of the following from the first principle:.
$\frac{5-x}{5+x}$
Let
$f(x)=\frac{5-x}{5+x}$
We need to find the derivative of f(x) i.e. f’(x)
We know that,
$\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}(\mathrm{x})}{\mathrm{h}}$ …(i)
$f(x)=\frac{5-x}{5+x}$
$\mathrm{f}(\mathrm{x}+\mathrm{h})=\frac{5-(\mathrm{x}+\mathrm{h})}{5+(\mathrm{x}+\mathrm{h})}=\frac{5-\mathrm{x}-\mathrm{h}}{5+\mathrm{x}+\mathrm{h}}$
Putting values in (i), we get
$\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\frac{5-\mathrm{x}-\mathrm{h}}{5+\mathrm{x}+\mathrm{h}}-\frac{5-\mathrm{x}}{5+\mathrm{x}}}{\mathrm{h}}$
$=\lim _{\mathrm{h} \rightarrow 0} \frac{\frac{(5-\mathrm{x}-\mathrm{h})(5+\mathrm{x})-(5-\mathrm{x})(5+\mathrm{x}+\mathrm{h})}{(5+\mathrm{x}+\mathrm{h})(5+\mathrm{x})}}{\mathrm{h}}$
$=\lim _{h \rightarrow 0} \frac{25+5 x-5 x-x^{2}-5 h-x h-\left[25+5 x+5 h-5 x-x^{2}-x h\right]}{h(5+x+h)(5+x)}$
$=\lim _{h \rightarrow 0} \frac{25-x^{2}-5 h-x h-25-5 h+x^{2}+x h}{h(5+x+h)(5+x)}$
$=\lim _{h \rightarrow 0} \frac{-10 h}{h(5+x+h)(5+x)}$
$=\lim _{h \rightarrow 0} \frac{-10}{(5+x+h)(5+x)}$
Putting h = 0, we get
$=\frac{-10}{(5+x+0)(5+x)}$
$=\frac{-10}{(5+x)(5+x)}$
$=\frac{-10}{(5+x)^{2}}$
Hence,
$f^{\prime}(x)=\frac{-10}{(5+x)^{2}}$
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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,