differmtiation

1. ASSIGNMENT 1

2. 1) If f(x) = x3 − 3x2 + 5, find f(0) + f(3).
Sol:
∴
f(0) = (0)3 − 3(0)2 +
5
by substituting x = 0
⟹ f(0) = 5
Also
,
f(3) = (3)3 − 3(3)2 +
5
by substituting x = 3
⟹ f(3) = 27 − 27 + 5
⟹ f(3) = 5
∴ f(0) + f(3) = 5 + 5
⟹ f(0) + f(3) = 10

3. 2) If f(x) = tan–1 x , find f(1)
4
Sol: Given: f(x) = tan–1 x
 f(1) = tan–1(1) by substituting x =1
⟹ f(1) =
4
4


4. Sol: Given: f(x) = x3 − 3x + sin x
Consider f(−x) = (−x)3 − 3(−x) + sin(−x)
⟹ f(−x) = − x3 + 3x − sin x
⟹ f(−x) = −{ x3 − 3x + sin x }
⟹ f(−x) = −f(x)
⟹ f(x) is an odd function.
3) State with proof whether the function f(x) = x3 − 3x +sinx
is even or odd

5. Sol : Given: f(x) = 3x4 + x2 + 5 − 3 cos x + 2sin2x …….. (i)
Consider f(−x) = 3(−x)4 + (−x)2 + 5 − 3 cos(−x) + 2sin2(−x)
= 3x4 + x2 + 5 − 3 cos x + 2sin2x
{sin(−θ) = − sin θ and cos (−θ) = cos θ}
⟹ f(− x) = f(x) from ( i)
4) If f(x) = 3x4 + x2 + 5 − 3 cos x + 2sin2x . Show that f(x) + f(−x) = 2f(x)

6. 5) If f(x) = = log(√1 + x2 + x) , find f(x) + f(− x)
Sol: Given: f(x) = log(√1 + x2 + x)
x
1
f(x) + f(− x) = log √1 + x + x + log(√1 +( - ) + (-x) )
⟹ f(x) + f(− x) = log √1 + + x + log √1 + x − x
⟹ f(x) + f(− x) = log √1 + x + x ∙ √1 + x − x
1 + (− x) + (− x)
𝑥2
⟹ f(x) + f(− x) = log 1 + x − x
⟹ (a + b)(a − b) = a − b
f(x) + f(− x) = log(1)
⟹ log + log
f(x) + f(− x) = 0

7. 6) Differentiate tan–1 x + cot–1 x w. r. t. x
Let y = tan–1 x + cot–1 x
y= 𝜋
2
{ tan–1 x + cot–1 x = }
Diff. w. r. t. x
dy/dx = 0⟹
⟹
𝜋
2

8. 7)Find dy/dx if y = 4 sin x − 8 cos x + tan x – sec x
Sol: Given: y = 4 sin x − 8 cos x + tan x − sec x
Diff. w. r. t. x
⟹ dy/dx = 4 cos x − 8(− sin x) + sec2 x − sec x tan x
⟹ dy/dx = 4 cos x + 8 sin x + sec2 x − sec x tan x

9. 8) Find dy/dx if y = x10 + 10x + ex
Sol : Given: y = 3x3 − 2ex + 4 sec x + 2√x
Diff. w. r. t. x
⟹dy/dx=10∙x
9
+10
x
∙log10+e
x

10. 9) Find dy/dx y = x . tanx
Sol: Given: y = x ∙ tan x
Diff. w. r. t. x
⟹ dy/dx = x ∙ sec2x + tan x ∙ 1

11. 10) Find dy/dx if y = (x + 1). log(x)
Sol : Given: y = (x + 1). log(x)
Diff. w. r. t. x
dy/dx = (x+1) . + log (x)
x
1