Learn preview of differential calculus and meaning of derivative of functions. Practice to double derivative, and standard trigonometric functions, chain rule and product rule derivative equation.
\(\Rightarrow\) \( \dfrac{dy}{dx}\) = Instantaneous rate of change of \(y\) with respect to \(x\)
\(\left[\because\, \lim \limits_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x}= \dfrac{dy}{dx}\right]\)
Then, average velocity = \(\dfrac{\Delta x}{\Delta t}\)
and, instantaneous velocity = \(\dfrac{d x}{d t}\)
\(a_{inst} = \dfrac{d}{dt}\left(v_{inst}\right)\)
\(a_{inst} = \dfrac{d \,v_{inst}}{dt}\)
A Velocity is a derivative of position with respect to time
B Velocity is a derivative of acceleration with respect to time
C Velocity is a derivative of position with respect to velocity
D Velocity is a derivative of acceleration with respect to position
\(y= f(x) = x^n\)
where 'n' is a real number
Then, the derivative of the function \(y= f(x)\) will be
\(\dfrac{dy}{dx} = n x^{n-1}\)
A \(3 t ^2 + 3\)
B \(9 t + 6\)
C \(10 t +5\)
D \( t + 3\)
A \(2\,e^x + 6\,x\)
B \(e^x + 3\,cos\,x\)
C \(2\,x + 4\,cos\,x + 5 \,e^x\)
D \(x + e^x\)
Then, by chain rule,
\(\dfrac{dy}{dt} =\dfrac{dy}{dx} . \dfrac{dx}{dt} \)
A \(e^{3x} . x\)
B \(e^{3x^2} (6\,x)\)
C \(2\,x\)
D \(e^x\)
A \(2 \, sin \,x + 5\; sec^2 \,x\)
B \(3 \, cos \,x + 6\, sin \,x\)
C \(2 \, cos \,x + 5 \;sec^2 \,x\)
D \(5 \, sec \,x \)
A \(2 \,e^x + x\)
B \(3 \,e^x + 2\,x\)
C \(3 \,e^x + 6\,x\)
D \(2 \,e^x + \dfrac{1}{x}\)
\(\dfrac{dy}{dx} = 0\)
\(v=\dfrac{dx}{dt}\)
When velocity (\(v\)) becomes zero, displacement \((x)\) is said to be maximum or minimum.
\(a = \dfrac{dv}{dt}\)
Here, when acceleration becomes zero, velocity is said to be maximum or minimum.
\(\dfrac{dy}{dx} = 0\)
and, \(\dfrac{d^2y}{dx^2} < 0\)
\(\dfrac{dy}{dx} = 0\)
and, \(\dfrac{d^2y}{dx^2} > 0\)
A Minima at \(x = \dfrac{-3}{2}\), Maxima does not exist
B Minima at \(x =0\), Maxima at \(x =3\)
C Minima at \(x =3\), Maxima at \(x =0\)
D Minima does not exist, Maxima at \(x = \dfrac{-3}{2}\)
Then, derivative of \(y= uv\) is given as
\(\dfrac{dy}{dx} = u\dfrac{dv}{dx} + v\dfrac{du}{dx}\)
A \(cos \,x + sin \,x\)
B \(3\,cos \,x +2\, sin \,x\)
C \(x\,cos \,x + sin \,x\)
D \( sin \,x\)
\(\dfrac{d^2y}{dx^2} = \dfrac{d}{dx} \left(\dfrac{dy}{dx}\right)\)
Instantaneous velocity, \(v= \dfrac{dx}{dt}\) and instantaneous acceleration, \(a= \dfrac{dv}{dt}\)
\(\Rightarrow a = \dfrac{d}{dt} \left(\dfrac{dx}{dt}\right)\)
\(\Rightarrow a = \dfrac{d^2x}{dt^2}\)