Proofs

Chapter 5 Class 12 Continuity and Differentiability
Concept wise

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Misc 21 (Method 1) If π¦ = |β( π(π₯) π(π₯) β(π₯)@π π π@π π Here ππ¦/ππ₯ = |β( πβ²(π₯) πβ²(π₯) ββ²(π₯)@π π π@π π π )| Expanding determinant ππ¦/ππ₯ = |πβ²(π₯)| |β 8(π&π@π&π)||βπβ²(π₯) | |β 8(π&π@π&π)||1+ ββ²(π₯) ||β 8(π&π@π&π)| ππ¦/ππ₯ = πβ²(π₯) (ππ βππ)βπβ²(π) (ππβππ) + ββ²(π) (ππβππ) ππ¦/ππ₯ = (ππ βππ) πβ²(π₯)β(ππβππ)πβ²(π₯) +(ππβππ) ββ²(π₯) Hence We need to prove that ππ/ππ = (ππ βππ) πβ²(π₯)β(ππβππ)πβ²(π₯) +(ππβππ) ββ²(π₯) Now, π¦ = |β( π(π₯) π(π₯) β(π₯)@π π π@π π π )| Expanding determinant π¦ = π(π₯)|β 8(π&π@π&π)|β π(π₯)|β 8(π&π@π&π)|+ β(π₯)|β 8(π&π@π&π)| π¦ = π(π₯) (ππ βππ)βπ(π) (ππβππ) + β(π) (ππβππ) π¦ = (ππ βππ) π(π₯)β(ππβππ)π(π₯)" +" (ππβππ) β(π₯)" " Differentiating π€.π.π‘.π₯. ππ¦/ππ₯ = π((ππ β ππ) π(π₯) β (ππ β ππ)π(π₯)" +" (ππ β ππ) β(π₯)" " )/ππ₯ ππ¦/ππ₯ = π((ππ β ππ) π(π₯))/ππ₯ β π((ππ β ππ)π(π₯))/ππ₯ + π((ππ β ππ) β(π₯))/ππ₯ ππ¦/ππ₯ = (ππβππ) π(π(π₯))/ππ₯ β (ππβππ) π(π(π₯))/ππ₯ + (ππβππ) π(β(π₯))/ππ₯ ππ¦/ππ₯ = (ππβππ) πβ²(π₯)β(ππβππ) πβ²(π₯) + (ππβππ) ββ²(π₯)" " Hence proved Misc 21 (Method 2) If π¦ = |β( π(π₯) π(π₯) β(π₯)@π π π@π π π )| , prove that ππ¦/ππ₯ = |β( πβ²(π₯) πβ²(π₯) ββ²(π₯)@π π π@π π π )| To Differentiate a determinant, We differentiate one row (or one column) at a time keeping others unchanged If π¦ = |β( π(π₯) π(π₯) β(π₯)@π π π@π π π )| ππ¦/ππ₯ = |β( πβ²(π₯) πβ²(π₯) ββ²(π₯)@π π π@π π π )| + |β(π(π₯) π(π₯) β(π₯)@(π)^β² (π)^β² (π)^β²@π π π )| + |β( π(π₯) π(π₯) β(π₯)@π π π@(π)β² (π)β² (π)β² )| ππ¦/ππ₯ = |β( πβ²(π₯) πβ²(π₯) ββ²(π₯)@π π π@π π π )| + |β(π(π₯) π(π₯) β(π₯)@0 0 0 @π π π )| + |β( π(π₯) π(π₯) β(π₯)@π π π@0 0 0 )| ππ¦/ππ₯ = |β( πβ²(π₯) πβ²(π₯) ββ²(π₯)@π π π@π π π )| + 0 + 0 ππ¦/ππ₯ = |β( πβ²(π₯) πβ²(π₯) ββ²(π₯)@π π π@π π π )| Hence proved. Using property If any one Row or column is 0 , then value of determinate is also 0 π )| , prove that ππ¦/ππ₯ = |β( πβ²(π₯) πβ²(π₯) ββ²(π₯)@π π π@π π π )|