Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $(a x+b)^{n}$
Let $f(x)=(a x+b)^{n}$. Accordingly, $f(x+h)=\{a(x+h)+b\}^{n}=(a x+a h+b)^{n}$
By first principle,
$f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f^{\prime}(x)}{h}$
$=\lim _{h \rightarrow 0} \frac{(a x+a h+b)^{n}-(a x+b)^{n}}{h}$
$=\lim _{h \rightarrow 0} \frac{(a x+b)^{n}\left(1+\frac{a h}{a x+b}\right)^{n}-(a x+b)^{n}}{h}$
$=(a x+b)^{n} \lim _{h \rightarrow 0} \frac{\left(1+\frac{a h}{a x+b}\right)^{n}-1}{h}$
$=(a x+b)^{n} \lim _{b \rightarrow 0} \frac{1}{n}\left[\left\{1+n\left(\frac{a h}{a x+b}\right)+\frac{n(n-1)}{\lfloor 2}\left(\frac{a h}{a x+b}\right)^{2}+\ldots\right\}-1\right]$
(Using binomial theorem)
$=(a x+b)^{n} \lim _{b \rightarrow 0} \frac{1}{h}\left[n\left(\frac{a h}{a x+b}\right)+\frac{n(n-1) a^{2} h^{2}}{\left\lfloor 2(a x+b)^{2}\right.}+\ldots(\right.$ Terms containing higher degrees of $\left.h)\right]$
$=(a x+b)^{n} \lim _{b \rightarrow 0}\left[\frac{n a}{(a x+b)}+\frac{n(n-1) a^{2} h}{\left\lfloor 2(a x+b)^{2}\right.}+\ldots\right]$
$=(a x+b)^{n}\left[\frac{n a}{(a x+b)}+0\right]$
$=n a \frac{(a x+b)^{a}}{(a x+b)}$
$=n a(a x+b)^{n-1}$
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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,