01) `\frac{d}{dx}` (`sec^{2}x - tan^{2}`x) = _________.
Answer: 0
02) `\frac{d}{dx}` (`sin^{-1}x + cos^{-1}`x) = _________.
Answer: 0
03) If y = log (sin `\frac{\pi}{2}`) then `\frac{d}{dx}` = __________.
Answer: cot`\frac{\pi}{2}`
04) `\frac{d}{dx}` `e^{-logx}` = _________.
Answer: -`\frac{e^{-logx}}{x}`
05) `\frac{d}{dx}` (`log_e` tanx) = ________.
Answer: 1
06) If y = `log_2 x`, then `\frac{d}{dx}`= __________.
Answer: `\frac{1}{x log2}`
07) If y = 4 root`\sqrt{x^3}` then `\frac{dy}{dx}` = _________.
Answer: `6x^5`
08) If y = `e^log` `e^{x^2}` then `\frac{dy}{dx}` = _________.
Answer: 2x
09) `\frac{dy}{dx}` `(y^4)` = _________.
Answer: `4y^3`
10)`\frac{dy}{dx}` y = `\sqrt{sinx}` = _________.
Answer: `\frac{cosx}{2\sqrt{sinx}}`
11) f(x) = `3x^2 - 2x + 1, then f'(-1)` = _________.
Answer: f'(x) = `6x^2 - 2`
12) `\frac{d}{dx}` (x • logx) = _________.
Answer: 1 + logx
13) `\frac{dy}{dx}` (`3sinx - 4sin^{3}x`) = _________.
Answer: `3cosx - 12sin^{2}x • cos x`
14) `\frac{dy}{dx}` `\frac{x}{cosx}` at x = 0 is _________.
Answer: 0
15) `\frac{dy}{dx}` `(sin x^0)` = __________.
Answer: `\frac{\pi}{180}` `cosx^0`
16) `\frac{dy}{dx}` (log cos x) = _________.
Answer: `\frac{-sinx}{log cosx}`
17) If f(x) = log `\sqrt{x^2 + 1}` then f'(0)` = _________.
Answer: 0
18) If `x^2 + y^2 = 29` then find `\frac{dy}{dx}` at point (2, 5) = ______.
Answer: 14
19) If f(x) = x sin x , then f'(0)` = __________.
Answer: 0
20) If f(x) = x cos x,then f'(0)` = __________.
Answer: 1
21) Differentiation `x^3 + 5x` with respect to x using definition.
Answer: `3x^2 + 5 `
22) Differentitaion `e^x` with respect to x using definition.
Answer: `e^x`
23) Differentitaion sin x with respect to x using first principle.
Answer: cosx
24) Find the `\frac{dy}{dx}` for y = `e^3x • cos2x`
Answer: `e^3x - sin2x + cos2x • e^3x `
25) Find the `\frac{dy}{dx}` for y = `e^x sinx cosx`.
Answer: `-e^x sin^2 x + e^x cos^2 x`
26) Differentiate sinx with respect to `tan^{2}x`.
Answer: ` 2sec^{2}x • tanx`
27) `\frac{d}{dx} (log_{3}x)`
Answer: `\frac{1}{x log_{e}3}`
28)`\frac{d}{dx} (sin x^0)`
Answer: `3sin^2x • cosx`
29) If y = `\frac{ax+b}{cx+d}`, then `\frac{dy}{dx}` = ___________.
Answer: `\frac{ad-bc}{(cx+d)_{2}}`
30)If x = `at_2` and y = 2at, then `\frac{dy}{dx}` = __________.
Answer: `\frac{1}{t}`
31) If y = `x_y` , then `\frac{dy}{dx}` = __________.
Answer: `\frac{y_2}{x(1- ylogx)}`
32)If x = a (1-cos `\theta`) and y = a (`\theta` - sin `\theta`), then `\frac{dy}{dx}` = __________.
Answer: `tan \frac{\theta}{2}`
33) If y = `e_x` sec x , then `\frac{dy}{dx}` = _____________.
Answer: `e_x secx tanx `
34)If `x^y = y^x , then \frac{dy}{dx}` = __________.
Answer: `\frac{y(x log y - y )}{x(ylogx - x)}`
35) If f(x) log `\sqrt{x^{2} + 1 }`, then f'(1) = __________.
Answer: `\frac{1}{2}`
36) If f(x) = `log_a x^n`, then f'(x) = ________.
Answer: `\frac{n}{x log_{e} a}`
37) If y =`tan^{-1} (cotx) + cot^{-1} (tanx) , then \frac{dy}{dx}` = ____________.
Answer: -2
38) If 2t = `v^2` then `\frac{dv}{dt}` = ____________.
Answer: `\frac{1}{v}`
39) If `\sqrt{x} + \sqrt{y} = \sqrt{a} , then \frac{dy}{dx}` = _________.
Answer: - `\sqrt{\frac{y}{x}}`
40) The equation of motion of a particle is s =`2t^3 - 9t^2 + 12t + 5.`then the velocity of the particle will be zero at t = __________.
Answer: (1, 2)
42) The equation of motion of a particle is s = `t^3 + 5t^2 + 7.` The velocity of the particle at t = 2 is __________ units.
Answer: 32
43) The equation of motion of a particle is s = `t^3 + 5t^2 + 3t + 5.` Then acceleration of the particle at t= 1 second is _____________ cm/`sec^2`.
Answer: -4
44) The maximum value of f(x) = sinx + cosx, x `\epsilon` ( 0, `\frac{\pi}{2})` is = __________.
Answer: `\sqrt{2}`
45) The maximun value of f(x) = `\frac{logx}{x}` (x > 0) is ___________.
Answer: `\frac{1}{e}`
46) The maximum value of f(x) = x + `\frac{1}{x}` is __________.
Answer: 2
47) The maximum value of f(x) = `x^2 - 4x + 2` is ________.
Answer: -1
48) The motion of particle is given by s = `t^3 - 6t^2 + 9t + 6`. Find the velocity of the particle when its acceleration is zero.
Answer: -3 cm/sec.
49) The equation of motion of a particle is s=`t^3 - 5t^2 + 3t +1 ` Find the time when the particle changes its direction.
Answer: t =`\frac{1}{3}` second
50) The equation of motion of a particle is s = `t^3 - 3t^2 + 4t + 3`. Find its velocity and acceleration at t = 2.
Answer: 6 cm/`sec^2`.
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