Question.
$\frac{a+b \sin x}{c+d \cos x}$
$\frac{a+b \sin x}{c+d \cos x}$
solution:
Given $y=\frac{a+b \sin x}{c+d \cos x}$
Applying division rule or quotient rule of differentiation that is
$\Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{t}}{\mathrm{y}}\right)=\frac{\mathrm{y} \cdot \mathrm{dt}}{\mathrm{dx}}-\mathrm{t} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$
$\Rightarrow y=\frac{a+b \sin x}{c+d \cos x}$
$\Rightarrow \frac{d y}{d x}=\frac{(c+d \cos x) \frac{d}{d x}(a+b \sin x)-(a+b \sin x) \frac{d}{d x}(c+d \cos x)}{(c+d \cos x)^{2}}$
On differentiating we get
$\Rightarrow \frac{d y}{d x}=\frac{(c+d \cos x)(b \cos x)-(a+b \sin x)(-d \sin x)}{(c+d \cos x)^{2}}$
$=\left[c b \cos x+b d \cos ^{2} x+a d \sin x+b d \sin ^{2} x\right] /(c+d \cos x)^{2}$
$=\left[\operatorname{cbcos} x+a d \sin x+b d\left(\cos ^{2} x+\sin ^{2} x\right)\right] /(c+d \cos x)^{2}$
$=[c b \cos x+a d \sin x+b d] /(c+d \cos x)^{2}$
Given $y=\frac{a+b \sin x}{c+d \cos x}$
Applying division rule or quotient rule of differentiation that is
$\Rightarrow \frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{t}}{\mathrm{y}}\right)=\frac{\mathrm{y} \cdot \mathrm{dt}}{\mathrm{dx}}-\mathrm{t} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}$
$\Rightarrow y=\frac{a+b \sin x}{c+d \cos x}$
$\Rightarrow \frac{d y}{d x}=\frac{(c+d \cos x) \frac{d}{d x}(a+b \sin x)-(a+b \sin x) \frac{d}{d x}(c+d \cos x)}{(c+d \cos x)^{2}}$
On differentiating we get
$\Rightarrow \frac{d y}{d x}=\frac{(c+d \cos x)(b \cos x)-(a+b \sin x)(-d \sin x)}{(c+d \cos x)^{2}}$
$=\left[c b \cos x+b d \cos ^{2} x+a d \sin x+b d \sin ^{2} x\right] /(c+d \cos x)^{2}$
$=\left[\operatorname{cbcos} x+a d \sin x+b d\left(\cos ^{2} x+\sin ^{2} x\right)\right] /(c+d \cos x)^{2}$
$=[c b \cos x+a d \sin x+b d] /(c+d \cos x)^{2}$
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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,
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