Students must practice these TS Inter 1st Year Maths 1B Study Material Chapter 9 Differentiation Ex 9(b) to find a better approach to solving the problems.
TS Inter 1st Year Maths 1B Differentiation 9(b)
I.
Question 1.
Find the derivatives of the following functions. (V.S.A.Q.)
(i) cotnx
Answer:
Let y = cotnx
Then \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = – n cotn – 1x (cosec2x)
= – n cotn-1 x (cosec2x)
(∵ \(\frac{\mathrm{d}}{\mathrm{dx}}\) (cot x) = – cosec2 x)
(ii) cosec4x
Answer:
Let y = cosec4x
Then \(\frac{d y}{d x}\) = 4 cosec3x \(\frac{\mathrm{d}}{\mathrm{d} x}\) (cosec x)
= 4 cosec3x (- cosec x cot x)
= – 4 cosec4x cot x
(iii) tan (ex)
Answer:
Let y = tan (ex)
Then \(\frac{d y}{d x}\) = sec2 (ex) \(\frac{d}{d x}\) (ex)
= ex.sec2(ex)
(iv) \(\frac{1-\cos 2 x}{1+\cos 2 x}\)
Answer:
Let y = \(\frac{1-\cos 2 x}{1+\cos 2 x}=\frac{2 \sin ^2 x}{2 \cos ^2 x}\) = tan2 x
∴ \(\frac{d y}{d x}\) = 2 tan x \(\frac{\mathrm{d}}{\mathrm{dx}}\) (tan x)
= 2 tan x sec2x
(v) sinmx cosnx
Answer:
Let y = sinmx cosnx
Then \(\frac{d y}{d x}\) = sinmx \(\frac{d}{d x}\) (cosnx) + cosn\(\frac{d y}{d x}\)(sinmx)
= sinmx n cosn – 1x (- sin x) + cosnx (m sinm – 1x) cos x
= m cosn + 1x sinm – 1x – n sinm + 1x cosn – 1x
(vi) sin mx . cos nx
Answer:
Let y = sin mx cos nx
Then \(\frac{d y}{d x}\) = sin mx . \(\frac{d}{d x}\) (cos nx) + cos nx \(\frac{d}{d x}\) (sin mx)
= sin mx (- n sin nx) + cos nx (m cos mx)
= – n sin mx sin nx + m cos nx cos mx
(vii) x . tan-1 x
Answer:
Let y = x . tan-1x
(viii) sin-1 (cos x)
Answer:
Let y = sin-1 (cos x)
(ix) log (tan 5x)
Answer:
Let y = log (tan 5x)
(x) sinh-1 \(\left(\frac{3 x}{4}\right)\)
Answer:
(xi) tan-1 (log x)
Answer:
Let y = tan-1 (log x)
(xii) log\(\left(\frac{x^2+x+2}{x^2-x+2}\right)\) (May 2006)
Answer:
(xiii) log [sin-1 (ex)]
Answer:
Let y = log [sin-1 (ex)]
(xiv) (sin x)2 (sin-1x)2
Answer:
Let y = (sin x)2 (sin-1x)2
(xv) \(\frac{\cos x}{\sin x+\cos x}\)
Answer:
(xvi) \(\frac{x\left(1+x^2\right)}{\sqrt{1-x^2}}\)
Answer:
(xvii) esin-1x
Answer:
Let y = esin-1x
(xviii) cos (log x + ex
Answer:
Let y = cos (log x + ex
\(\frac{d y}{d x}\) = – sin(log x + ex) \(\frac{\mathrm{d}}{\mathrm{dx}}\) (log x + ex)
= – sin (log x + ex) (\(\frac{1}{x}\) + ex)
(xix) \(\frac{\sin (x+a)}{\cos x}\)
Answer:
(xx) cot-1 (cosec 3x)
Answer:
Let y = cot-1 (cosec 3x)
Question 2.
Find the derivatives of the following functions. (V.S.A.Q.)
(i) x = sin h2y
Answer:
(ii) x = tanh2y
Answer:
(iii) x = esinh y
Answer:
\(\frac{\mathrm{dx}}{\mathrm{dy}}\) = esin hy \(\frac{\mathrm{d}}{\mathrm{dy}}\) (sin hy)
= esinhy cos hy = x cos hy
(iv) x tan (e-y)
Answer:
\(\frac{\mathrm{dx}}{\mathrm{dy}}\) = sec2(e-y) \(\frac{d}{d y}\) (e-y)
= – sec2(e-y) (e-y)
= – e-y (1 + tan2 (e-y)) = – e-y(1 + x2)
(v) x = log (1 + sin2y)
Answer:
(vi) x = log(1 + √y)
Answer:
1 + √y = ex
√y = ex – 1
y = (ex – 1)2
∴ \(\frac{\mathrm{dy}}{\mathrm{dx}}\) = 2(ex – 1) . \(\frac{\mathrm{d}}{\mathrm{dx}}\) (ex
= 2(ex – 1) ex
= 2√y (√y + 1) = 2(y + √y)
II. Find the derivatives of the following functions. (V.S.A.Q.)
(i) cos [log (cot x)]
Answer:
y = cos [log (cot x)]
Let cot x = u, log u = v, so that y = cos v
(ii) sin h-1
Answer:
(iii) log [cot (1 – x2)]
Answer:
y = log [cot (1 – x2)]
Let 1 – x2 = u, cot u = v, and y = log v
(iv) sin [cos (x2)]
Answer:
y = sin [cos (x2)]
Let x2 = u, v = cos u and y = sin v
\(\frac{d y}{d x}=\frac{d y}{d v} \cdot \frac{d v}{d u} \cdot \frac{d u}{d x}\)
= cos v(- sin u) (2x)
= – 2x cos (cos u) sin u
= – 2x cos [(cos (x2)] sin (x2)
(v) sin [tan-1 (ex)]
Answer:
y = sin [tan-1 (ex)]
Let ex = u, tan-1u = v and y = sin v
(vi) \(\frac{\sin (a x+b)}{\cos (c x+d)}\)
Answer:
(vii) tan-1 (tanh \(\frac{x}{2}\))
Answer:
(viii) sin x (tan-1x)
Answer:
Let y = sin x (tan-1x)2
III. Find the derivatives of the following functions.
Question 1.
sin-1\(\left(\frac{b+a \sin x}{a+b \sin x}\right)\) (a > o, b > 0) (E.Q.)
Answer:
Question 2.
cos-1\(\left(\frac{b+a \cos x}{a+b \cos x}\right)\) (a > 0, b > 0) (E.Q.)
Answer:
Question 3.
tan-1 \(\left(\frac{\cos x}{1+\cos x}\right)\)
Answer: