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dy/dx=(6x+1)/(√(4x+1)), Calculus










































Given that y=x√(4x+1)
Show that dy/dx=(6x+1)/(√(4x+1))

Comments for dy/dx=(6x+1)/(√(4x+1)), Calculus

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Nov 28, 2011
Derivatives
by: Staff

Question:

Given that y=x√(4x+1)
Show that dy/dx=(6x+1)/(√(4x+1))


Answer:


Begin by applying the Product Rule (also called Leibniz's Law) to your problem:

f(x) = U*V

f’(x) = U*V’ + V*U’

Therefore:

y = x * √(4x+1)

dy/dx = x * dy/dx(√(4x+1)) + (√(4x+1)) * dy/dx(x)



Differentiate each part of the expression

dy/dx(√(4x+1))

= dy/dx (4x+1)^.5

= 2/[(√(4x+1)]


dy/dx(x)

= 1


dy/dx = x * dy/dx(√(4x+1)) + (√(4x+1)) * dy/dx(x)

= x * 2/[(√(4x+1)] + (√(4x+1)) * 1


= 2x/[(√(4x+1)] + √(4x+1)


dy/dx = 2x/[(√(4x+1)] + √(4x+1)


Combine the two terms by converting the second term to a fraction, and then adding the first and second terms.


Ensure both terms are fractions with a common denominator.

This can be accomplished by multiplying the second term by [√(4x+1)]/[√(4x+1)].


dy/dx = 2x/[(√(4x+1)] + [√(4x+1)]*{[√(4x+1)]/[√(4x+1)]}

= 2x/[(√(4x+1)] + [√(4x+1)]*{[√(4x+1)]/[√(4x+1)]}

= 2x/[(√(4x+1)] + (4x+1)/[√(4x+1)]

= [2x + (4x + 1)]/[(√(4x+1)]

= [2x + 4x + 1]/[(√(4x+1)]

= [6x + 1]/[(√(4x+1)]


The final answer is: dy/dx = (6x + 1)/((√(4x+1))



Thanks for writing.

Staff
www.solving-math-problems.com



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