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Continuity And Differentiability

Question

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Differentiate : (x + 3)² (x + 4)³ (x + 5)^4.w.r.t.x.

Solution

Let space space space space space space space space space space straight y equals left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent left parenthesis straight x plus 5 right parenthesis to the power of 4 therefore space space space space space log space straight y equals log left square bracket left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent left parenthesis straight x plus 5 right parenthesis to the power of 4 right square bracket therefore space space space space space space space space space space space space space space equals log left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent log left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent log left parenthesis straight x plus 5 right parenthesis to the power of 4 therefore space space space space space log space straight y equals 2 log left parenthesis straight x plus 3 right parenthesis plus 3 log left parenthesis straight x plus 4 right parenthesis plus 4 to the power of space log left parenthesis straight x plus 5 right parenthesis therefore 1 over straight y dy over dx equals fraction numerator 2 over denominator straight x plus 3 end fraction plus fraction numerator 3 over denominator straight x plus 4 end fraction plus fraction numerator 4 over denominator straight x plus 5 end fraction therefore space space space space space space dy over dx equals left parenthesis straight x plus 3 right parenthesis to the power of 2 space end exponent left parenthesis straight x plus 4 right parenthesis to the power of 3 space end exponent left parenthesis straight x plus 5 right parenthesis to the power of 4 open square brackets fraction numerator 2 over denominator straight x plus 3 end fraction plus fraction numerator 3 over denominator straight x plus 4 end fraction plus fraction numerator 4 over denominator straight x plus 5 end fraction close square brackets

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