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प्रश्न
Solve the following :
The values of f(x), g(x), f'(x) and g'(x) are given in the following table :
| x | f(x) | g(x) | f'(x) | fg'(x) |
| – 1 | 3 | 2 | – 3 | 4 |
| 2 | 2 | – 1 | – 5 | – 4 |
Match the following :
| A Group – Function | B Group – Derivative |
| (A)`"d"/"dx"[f(g(x))]"at" x = -1` | 1. – 16 |
| (B)`"d"/"dx"[g(f(x) - 1)]"at" x = -1` | 2. 20 |
| (C)`"d"/"dx"[f(f(x) - 3)]"at" x = 2` | 3. – 20 |
| (D)`"d"/"dx"[g(g(x))]"at"x = 2` | 5. 12 |
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उत्तर
(A) `"d"/"dx"[f(g(x))]`
= `f'(g(x))."d"/"dx"(g(x))`
= f'(g(x)) xg'(x)
∴ `"d"/"dx"[f(g(x))]` at x = – 1
= f'(g(– 1)) x g'(– 1)
= f'(2) x g'(– 1) ...[∵ g(x) = 2, when x = – 1]
= – 5 x 4
= – 20
(B) `"d"/"dx"[g(f(x) - 1)]`
= `g'(f(x) - 1)."d"/"dx"[f(x) - 1]`
= g'(f(x) – 1) x [f'(x) – 0]
∴ `"d"/"dx"[gf(x) - 1]` at x = – 1
= g'(f(– 1)– 1) xx f'( –1)
= g'(3 – 1) x f'(– 1) ...[∵ f(x) 33, when x = – 1]
= g'(2) x f'(– 1)
= (– 4)(– 3)
= 12
(C) `"d"/"dx"[f(f(x) - 3)]`
= `f'(f(x) - 3)."d"/"dx"[f(x) - 3]`
= f'(f(x) – 3) x [f'(x) – 0]
∴ `"d"/"dx"[f(f(x) - 3)]` at = 2
= f"(f(2) – 3) x f'(2)
= f'(2 – 3) x f'(2) ...[∵ f(x) = 2, when x = 2]
= f'(– 1) x f'(2)
= (– 3)(– 5)
= 15
(D) `"d"/"dx"[g(g(x))]`
= `g'(g(x))."d"/"dx"[g(x)]`
= g'(g(x)) x g'(x)
∴ `"d"/"dx"[g(g(x))]`at x = 2
= g'(g(2)) x g'(2)
= g'(– 1) c g'(2) ...[∵ g(x) = – 1at x = 2]
= 4(– 4)
= – 16
Hence, (A) →3, (B) → 5, (C) → 4, (D) → 1.
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