Question

$If space straight x to the power of straight x plus straight x to the power of straight y plus straight y to the power of straight x space equals space straight a to the power of straight b comma space space then space find space dy over dx.$

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Solution

straight x to the power of straight x plus straight x to the power of straight y plus straight y to the power of straight x space equals straight a to the power of straight b space space space... left parenthesis straight i right parenthesis Let space straight u space equals space straight x to the power of straight x log space straight u space equals space straight x space log space straight x 1 over straight u. du over dx equals straight x.1 over straight x plus log space straight x therefore space space du over dx equals straight x to the power of straight x left parenthesis 1 plus log space straight x right parenthesis Let space straight v equals space straight x to the power of straight y log space straight v space equals space straight y space log space straight x 1 over straight v. dv over dx space equals space open parentheses straight y over straight x plus log space straight x dy over dx close parentheses therefore space dv over dx equals straight x to the power of straight y open parentheses straight y over straight x plus logx dy over dx close parentheses Let space straight w space equals straight y to the power of straight x Log space straight w space equals space straight x space log space straight y 1 over straight w. dw over dx equals space open parentheses straight x over straight y. dy over dx plus log space straight y close parentheses therefore space dw over dx equals straight y to the power of straight x open parentheses log space straight y space plus space straight x over straight y. dy over dx close parentheses

left parenthesis straight i right parenthesis space can space be space writeen space as straight u plus straight v plus straight w space equals space straight a to the power of straight b du over dx plus dv over dx plus dw over dx equals 0 rightwards double arrow space space straight x to the power of straight x left parenthesis 1 plus logx right parenthesis plus straight x to the power of straight y open parentheses straight y over straight x plus logx dy over dx close parentheses plus straight y to the power of straight x open parentheses logy plus straight x over straight y dy over dx close parentheses space equals space 0 rightwards double arrow space space straight x to the power of straight x plus straight x to the power of straight x logx plus straight x to the power of straight y. straight y over straight x plus straight x to the power of straight y. logx. dy over dx plus straight y to the power of straight x. logy plus straight y to the power of straight x. straight x over straight y. dy over dx equals 0 rightwards double arrow space space dy over dx open parentheses straight x to the power of straight y. logx space plus space straight y to the power of straight x. straight x over straight y close parentheses space equals space straight x to the power of straight x plus straight x to the power of straight x logx plus straight x to the power of straight y. straight y over straight x plus straight y to the power of straight x. log space straight y rightwards double arrow dy over dx open parentheses straight x to the power of straight y. logx space plus space xy to the power of straight x minus 1 end exponent close parentheses space equals space open parentheses straight x to the power of straight x plus straight x to the power of straight x logx plus yx to the power of straight y minus 1 end exponent plus straight y to the power of straight x. log space straight y close parentheses therefore space space dy over dx equals fraction numerator left parenthesis straight x to the power of straight x plus straight x to the power of straight x logx space plus yx to the power of straight y minus 1 end exponent plus straight y to the power of straight x. log space straight y right parenthesis over denominator left parenthesis straight x to the power of straight y. logx plus xy to the power of straight x minus 1 end exponent right parenthesis end fraction

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Vector Algebra

$If space straight x to the power of straight x plus straight x to the power of straight y plus straight y to the power of straight x space equals space straight a to the power of straight b comma space space then space find space dy over dx.$

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0

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For Daily Free Study Material Join wiredfaculty

Vector Algebra

Some More Questions From Vector Algebra Chapter

Determine order and degree (if defined) of differential equations:y' + y = ex

Determine order and degree (if defined) of differential equations:y'' + (y')2 + 2 y = 0

Determine order and degree (if defined) of differential equations:y″ + 2y′ + sin y = 0

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Determine order and degree (if defined) of differential equations:
y' + y = e^x

Determine order and degree (if defined) of differential equations:
y'' + (y')² + 2 y = 0

Determine order and degree (if defined) of differential equations:
y″ + 2y′ + sin y = 0