Differentiate:.
$\left(x^{3} \cos x-2^{x} \tan x\right)$
To find: Differentiation of $\left(x^{3} \cos x-2^{x} \tan x\right)$
Formula used: (i) (uv)' $=u^{\prime} v+u v^{\prime}$ (Leibnitz or product rule)
(ii)
$\frac{d x^{n}}{d x}=n x^{n-1}$
(iii)
$\frac{d \cos x}{d x}=-\sin x$
(iv)
$\frac{d a^{x}}{d x}=a^{x} \log a$
(v)
$\frac{d \tan x}{d x}=\sec ^{2} x$
Here we have two function $\left(x^{3} \cos x\right)$ and $\left(2^{x} \tan x\right)$
We have two differentiate them separately
Let us assume $g(x)=\left(x^{3} \cos x\right)$
And $h(x)=\left(2^{x} \tan x\right)$
Therefore, f(x) = g(x) – h(x)
$\Rightarrow f^{\prime}(x)=g^{\prime}(x)-h^{\prime}(x) \ldots$ (i)
Applying product rule on g(x)
Let us take $u=x^{3}$ and $v=\cos x$
$u^{\prime}=\frac{d u}{d x}=\frac{d\left(x^{3}\right)}{d x}=3 x^{2}$
$v^{\prime}=\frac{d v}{d x}=\frac{d(\cos x)}{d x}=-\sin x$
Putting the above obtained values in the formula:-
$(u v)^{\prime}=u^{\prime} v+u v^{\prime}$
$\left[x^{3} \cos x\right]^{\prime}=3 x^{2} \times \cos x+x^{3} \times-\sin x$
$=3 x^{2} \cos x-x^{3} \sin x$
$=x^{2}(3 \cos x-x \sin x)$
$g^{\prime}(x)=x^{2}(3 \cos x-x \sin x)$
Applying product rule on $\mathrm{h}(\mathrm{x})$
Let us take $u=2^{x}$ and $v=\tan x$
$u^{\prime}=\frac{d u}{d x}=\frac{d\left(2^{x}\right)}{d x}=2^{x} \log 2$
$v^{\prime}=\frac{d v}{d x}=\frac{d(\tan x)}{d x}=\sec ^{2} x$
Putting the above obtained values in the formula:-
$(u v)^{\prime}=u^{\prime} v+u v^{\prime}$
$\left[2^{x} \tan x\right]^{\prime}=2^{x} \log 2 x \tan x+2^{x} \times \sec ^{2} x$
$=2^{x}\left(\log 2 \tan x+\sec ^{2} x\right)$
$h^{\prime}(x)=2^{x}\left(\log 2 \tan x+\sec ^{2} x\right)$
Putting the above obtained values in eqn. (i)
$f^{\prime}(x)=x^{2}(3 \cos x-x \sin x)-2^{x}\left(\log 2 \tan x+\sec ^{2} x\right)$
Ans) $x^{2}(3 \cos x-x \sin x)-2^{x}\left(\log 2 \tan x+\sec ^{2} x\right)$
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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,