List of top Mathematics Questions on Applications of Derivatives asked in UP Board XII

If \( x = a(0 - \sin \theta) \), \( y = a(1 + \cos \theta) \), find \[ \frac{dy}{dx}. \] 

Find the least value of ‘a’ for which the function \( f(x) = x^2 + ax + 1 \) is increasing on the interval \( [1, 2] \).

Find the interval in which the function \[ f(x) = 4x^3 - 6x^2 - 72x + 30 \] is (i) increasing and (ii) decreasing.
Find the equation of tangent at $t = \dfrac{\pi}{2}$ on the curve $x = a \sin^{3}t$, $y = b \cos^{3}t$.
Prove that the function \[ f(x) = \tan^{-1} (\sin x + \cos x), x > 0 \text{ is always increasing function on } (0, \frac{\pi}{4}). \]
Find the equation of tangent at the point \( (am^2, am^3) \) on the curve \[ ay^2 = x^3. \]
Find those points on the curve \[ \frac{x^2}{4} + \frac{y^2}{25} = 1, \] on which the normal is parallel to the x-axis.
If the length of a rectangle is decreasing at the rate of 3 cm/min and width is increasing at the rate of 2 cm/min, then find the rate of change in perimeter of the rectangle when \( x = 10 \) cm and \( y = 6 \) cm, where \( x \) = length and \( y \) = width.
Find the intervals in which the function $f(x)=\sin x + \cos x,\; 0\leq x \leq 2\pi$ is increasing or decreasing.
Find the point on the curve \[ y = x^3 - 11x + 5 \] at which the tangent is \[ y = x - 11. \]
Find the equation of tangent of the curve $y = \cos(x+y)$, $-2\pi \leq x \leq 2\pi$, which is parallel to the line $x + 2y = 0$.
The slope of the normal to the curve $y = 2x^{2} + 3\sin x$ at $x = 0$ will be: