If y = Determinant | f(x) g(x) h(x) l m n a b c|, prove dy/dx = | f'(x (2025)

Chapter 5 Class 12 Continuity and Differentiability

Serial order wise

• Ex 5.1
• Ex 5.2
• Ex 5.3
• Ex 5.4
• Ex 5.5
• Ex 5.6
• Ex 5.7
• Examples

• Miscellaneous

• Case Based Questions (MCQ)
• NCERT Exemplar - MCQs
• Rolle's and Mean Value Theorem

Next: Misc 22 Important → Go Ad-free

Transcript

Misc 21 (Method 1) If 𝑦 = |█( 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 Here 𝑑𝑦/𝑑𝑥 = |█( 𝑓′(𝑥) 𝑔′(𝑥) ℎ′(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )|Expanding determinant 𝑑𝑦/𝑑𝑥 = |𝑓′(𝑥)| |■8(𝑚&𝑛@𝑏&𝑐)||−𝑔′(𝑥) | |■8(𝑙&𝑛@𝑎&𝑐)||1+ ℎ′(𝑥) ||■8(𝑙&𝑚@𝑎&𝑏)|𝑑𝑦/𝑑𝑥 = 𝑓′(𝑥) (𝑚𝑐 −𝑏𝑛)−𝑔′(𝑛) (𝑙𝑐−𝑎𝑛) + ℎ′(𝑛) (𝑙𝑏−𝑎𝑚)𝑑𝑦/𝑑𝑥 = (𝑚𝑐 −𝑏𝑛) 𝑓′(𝑥)−(𝑙𝑐−𝑎𝑛)𝑔′(𝑥) +(𝑙𝑏−𝑎𝑚) ℎ′(𝑥) Hence We need to prove that𝒅𝒚/𝒅𝒙 = (𝑚𝑐 −𝑏𝑛) 𝑓′(𝑥)−(𝑙𝑐−𝑎𝑛)𝑔′(𝑥) +(𝑙𝑏−𝑎𝑚) ℎ′(𝑥)Now,𝑦 = |█( 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )|Expanding determinant𝑦 = 𝑓(𝑥)|■8(𝑚&𝑛@𝑏&𝑐)|− 𝑔(𝑥)|■8(𝑙&𝑛@𝑎&𝑐)|+ ℎ(𝑥)|■8(𝑙&𝑚@𝑎&𝑏)|𝑦 = 𝑓(𝑥) (𝑚𝑐 −𝑏𝑛)−𝑔(𝑛) (𝑙𝑐−𝑎𝑛) + ℎ(𝑛) (𝑙𝑏−𝑎𝑚) 𝑦 = (𝑚𝑐 −𝑏𝑛) 𝑓(𝑥)−(𝑙𝑐−𝑎𝑛)𝑔(𝑥)" +" (𝑙𝑏−𝑎𝑚) ℎ(𝑥)" "Differentiating 𝑤.𝑟.𝑡.𝑥. 𝑑𝑦/𝑑𝑥 = 𝑑((𝑚𝑐 − 𝑏𝑛) 𝑓(𝑥) − (𝑙𝑐 − 𝑎𝑛)𝑔(𝑥)" +" (𝑙𝑏 − 𝑎𝑚) ℎ(𝑥)" " )/𝑑𝑥 𝑑𝑦/𝑑𝑥 = 𝑑((𝑚𝑐 − 𝑏𝑛) 𝑓(𝑥))/𝑑𝑥 − 𝑑((𝑙𝑐 − 𝑎𝑛)𝑔(𝑥))/𝑑𝑥 + 𝑑((𝑙𝑏 − 𝑎𝑚) ℎ(𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = (𝑚𝑐−𝑏𝑛) 𝑑(𝑓(𝑥))/𝑑𝑥 − (𝑙𝑐−𝑎𝑛) 𝑑(𝑔(𝑥))/𝑑𝑥 + (𝑙𝑏−𝑎𝑚) 𝑑(ℎ(𝑥))/𝑑𝑥 𝑑𝑦/𝑑𝑥 = (𝑚𝑐−𝑏𝑛) 𝑓′(𝑥)−(𝑙𝑐−𝑎𝑛) 𝑔′(𝑥) + (𝑙𝑏−𝑎𝑚) ℎ′(𝑥)" "Hence provedMisc 21 (Method 2) If 𝑦 = |█( 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )| , prove that 𝑑𝑦/𝑑𝑥 = |█( 𝑓′(𝑥) 𝑔′(𝑥) ℎ′(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )| To Differentiate a determinant,We differentiate one row (or one column) at a time keeping others unchanged If 𝑦 = |█( 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )| 𝑑𝑦/𝑑𝑥 = |█( 𝑓′(𝑥) 𝑔′(𝑥) ℎ′(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )| + |█(𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@(𝑙)^′ (𝑚)^′ (𝑛)^′@𝑎 𝑏 𝑐 )| + |█( 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@𝑙 𝑚 𝑛@(𝑎)′ (𝑏)′ (𝑐)′ )|𝑑𝑦/𝑑𝑥 = |█( 𝑓′(𝑥) 𝑔′(𝑥) ℎ′(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )| + |█(𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@0 0 0 @𝑎 𝑏 𝑐 )| + |█( 𝑓(𝑥) 𝑔(𝑥) ℎ(𝑥)@𝑙 𝑚 𝑛@0 0 0 )|𝑑𝑦/𝑑𝑥 = |█( 𝑓′(𝑥) 𝑔′(𝑥) ℎ′(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )| + 0 + 0 𝑑𝑦/𝑑𝑥 = |█( 𝑓′(𝑥) 𝑔′(𝑥) ℎ′(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )|Hence proved. Using property If any one Row or column is 0 , then value of determinate is also 0𝑐 )| , prove that 𝑑𝑦/𝑑𝑥 = |█( 𝑓′(𝑥) 𝑔′(𝑥) ℎ′(𝑥)@𝑙 𝑚 𝑛@𝑎 𝑏 𝑐 )|