All answers apply on the domain of the original function. Preserve excluded denominator zeros after simplification. Square-root derivatives below require x > 0. In Q5–Q6, n is a positive integer. For integer exponents equal to zero in later questions, differentiate the constant factor directly before using a factored answer; this avoids introducing a spurious division by zero.
Q5. Find the derivative of xn + axn−1 + a2xn−2 + … + an−1x + an for a fixed real number a (NCERT Exercise 12.2, Q6).
Term by term: nxn−1 + (n − 1)axn−2 + (n − 2)a2xn−3 + … + an−1.
Q6. Find the derivative of (xn − an)/(x − a) for a constant a (NCERT Exercise 12.2, Q8).
By the quotient rule, (nxn−1(x − a) − (xn − an))/
Q7. Find the derivatives of (i) 2x − 3/4, (ii) (5x3 + 3x − 1)(x − 1), (iii) x−3(5 + 3x) (NCERT Exercise 12.2, Q9 (i)–(iii)).
(i) 2. (ii) Expanding, 5x4 − 5x3 + 3x2 − 4x + 1, so 20x3 − 15x2 + 6x − 4. (iii) 5x−3 + 3x−2 gives −15x−4 − 6x−3.
Q8. Find the derivatives of (iv) x5(3 − 6x−9), (v) x−4(3 − 4x−5), (vi) 2/(x + 1) − x2/(3x − 1) (NCERT Exercise 12.2, Q9 (iv)–(vi)).
(iv) 3x5 − 6x−4 gives 15x4 + 24x−5. (v) 3x−4 − 4x−9 gives −12x−5 + 36x−10. (vi) −2/(x + 1)2 − x(3x − 2)/(3x − 1)2.
Q9. Find the derivatives of sin x cos x, sec x, 5 sec x + 4 cos x and cosec x (NCERT Exercise 12.2, Q11 (i)–(iv)).
(i) cos2 x − sin2 x = cos 2x.
(ii) sec x tan x.
(iii) 5 sec x tan x − 4 sin x.
(iv) −cosec x cot x.
Q10. Find the derivatives of 3 cot x + 5 cosec x, 5 sin x − 6 cos x + 7 and 2 tan x − 7 sec x (NCERT Exercise 12.2, Q11 (v)–(vii)).
(v) −3 cosec2 x − 5 cosec x cot x.
(vi) 5 cos x + 6 sin x.
(vii) 2 sec2 x − 7 sec x tan x.
Q11. From first principle, find the derivatives of x + 1/x, sin x + cos x and x sin x (NCERT Examples 19 (ii) and 20, pages 249–251).
For x + 1/x: the quotient is [h − h/(x(x + h))]/h = 1 − 1/(x(x + h)) → 1 − 1/x2.
For sin x + cos x: expanding, the quotient is
(sin h/h)(cos x − sin x)
+ (sin x + cos x)(cos h − 1)/h,
which tends to cos x − sin x.
For x sin x: the quotient is x(sin(x + h) − sin x)/h + sin(x + h), which tends to x cos x + sin x.
In Example 20 (i), NCERT’s first line (page 250) leaves out the “lim” before the fraction.
Q12. From first principle, find the derivatives of −x, (−x)−1, sin(x + 1) and cos(x − π/8) (NCERT Miscellaneous Exercise on Chapter 12, Q1).
For −x: (−(x + h) + x)/h = −1.
For (−x)−1 = −1/x: (−1/(x + h) + 1/x)/h = 1/(x(x + h)) → 1/x2.
For sin(x + 1): the quotient is cos(x + 1 + h/2) × [sin(h/2)/(h/2)] → cos(x + 1).
In the same way, cos(x − π/8) gives −sin(x − π/8).
Q13. Differentiate x + a, (px + q)(r/x + s) and (ax + b)(cx + d)2 (NCERT Miscellaneous Exercise, Q2–Q4). In Q13 to Q22, as NCERT says, a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers.
NCERT Q2: 1.
NCERT Q3: ps − qr/x2.
NCERT Q4: a(cx + d)2 + 2c(ax + b)(cx + d).
Q14. Differentiate (ax + b)/(cx + d), (1 + 1/x)/(1 − 1/x) and 1/(ax2 + bx + c) (NCERT Miscellaneous Exercise, Q5–Q7).
NCERT Q5: (ad − bc)/(cx + d)2.
NCERT Q6: (1 + 1/x)/(1 − 1/x) = (x + 1)/(x − 1) for x ≠ 0, 1, so −2/(x − 1)2.
NCERT Q7: −(2ax + b)/(ax2 + bx + c)2.
Q15. Differentiate (ax + b)/(px2 + qx + r), (px2 + qx + r)/(ax + b) and a/x4 − b/x2 + cos x (NCERT Miscellaneous Exercise, Q8–Q10).
NCERT Q8: (−apx2 − 2bpx + ar − bq)/
NCERT Q9: (apx2 + 2bpx + bq − ar)/
NCERT Q10: −4a/x5 + 2b/x3 − sin x.
Q16. Differentiate 4√x − 2, (ax + b)n and (ax + b)n(cx + d)m (NCERT Miscellaneous Exercise, Q11–Q13).
NCERT Q11: 2/√x, for x > 0; there is no finite right derivative at 0.
NCERT Q12: na(ax + b)n−1.
NCERT Q13: (ax + b)n−1(cx + d)m−1
× [na(cx + d) + mc(ax + b)].
The chain rule is a Class 12 topic; here (ax + b)n can be done by induction on n with the product rule for positive n, as NCERT proves (xn)′; for negative n, write it as 1/(ax + b)|n| and use the quotient rule.
Q17. Differentiate sin(x + a), cosec x cot x and cos x/(1 + sin x) (NCERT Miscellaneous Exercise, Q14–Q16).
NCERT Q14: cos(x + a), from sin x cos a + cos x sin a.
NCERT Q15: −cosec x(cot2 x + cosec2 x).
NCERT Q16: −1/(1 + sin x).
Q18. Differentiate (sin x + cos x)/(sin x − cos x), (sec x − 1)/(sec x + 1) and sinn x (NCERT Miscellaneous Exercise, Q17–Q19).
NCERT Q17: −2/(sin x − cos x)2.
NCERT Q18: 2 sec x tan x/(sec x + 1)2.
NCERT Q19: n sinn−1 x cos x (by induction with the product rule for positive n; for negative n, use the quotient rule on 1/sin|n| x).
Q19. Differentiate (a + b sin x)/(c + d cos x), sin(x + a)/cos x and x4(5 sin x − 3 cos x) (NCERT Miscellaneous Exercise, Q20–Q22).
NCERT Q20: (bc cos x + ad sin x + bd)/
NCERT Q21: cos a/cos2 x.
NCERT Q22: 4x3(5 sin x − 3 cos x) + x4(5 cos x + 3 sin x).
Q20. Differentiate (x2 + 1) cos x, (ax2 + sin x)(p + q cos x) and (x + cos x)(x − tan x) (NCERT Miscellaneous Exercise, Q23–Q25).
NCERT Q23: 2x cos x − (x2 + 1) sin x.
NCERT Q24: (2ax + cos x)(p + q cos x) − q sin x(ax2 + sin x).
NCERT Q25: (1 − sin x)(x − tan x) + (x + cos x)(1 − sec2 x).
Q21. Differentiate (4x + 5 sin x)/(3x + 7 cos x) and x2 cos(π/4)/sin x (NCERT Miscellaneous Exercise, Q26 and Q27).
NCERT Q26: the numerator is 15x cos x + 28 cos x + 28x sin x − 15 sin x + 35, and the denominator is (3x + 7 cos x)2.
NCERT Q27: cos(π/4) is a constant, so x(2 sin x − x cos x)/(√2 sin2 x).
Q22. Differentiate x/(1 + tan x), (x + sec x)(x − tan x) and x/sinn x (NCERT Miscellaneous Exercise, Q28–Q30).
NCERT Q28: (1 + tan x − x sec2 x)/
NCERT Q29: (1 + sec x tan x)(x − tan x) + (x + sec x)(1 − sec2 x).
NCERT Q30: (sin x − nx cos x)/sinn+1 x.