NCERT Class 12 · Ch 5 Exercise 5.5 · Q18
NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 18
Show (uvw)′=u′vw+uv′w+uvw′ by repeated product rule and logarithmic differentiation.
💡 Concept Hints
- Identify the domain and the inner and outer functions, or the relevant one-sided limits.
- Apply the appropriate differentiation or limit rule, keep all chain factors, and check exceptional points.
✅ Step-by-Step Solution
Final Answer: (uvw)′=u′vw+uv′w+uvw′.
- Repeated product rule: (uv·w)′=(u′v+uv′)w+uvw′, yielding all three terms.
- Where u,v,w≠0, log|uvw|=log|u|+log|v|+log|w|, so (uvw)′/(uvw)=u′/u+v′/v+w′/w. Multiplication gives the same identity.
- At zeros the logarithmic intermediate is unavailable, but the direct product-rule proof still applies.
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