Find the derivative of #sqrt(3x-5)# using the definition of the derivative. Help?

1 Answer
Jim G.
May 21, 2018

#3/(2sqrt(3x-5))#

Explanation:

#"using differentiation from first principles"#

#•color(white)(x)f'(x)=lim_(hto0)(f(x+h)-f(x))/h#

#rArrf'(x)=lim_(hto0)(sqrt(3(x+h)-5)-sqrt(3x-5))/h#

#"multiply numerator/denominator by the conjugate of "#
#"the numerator"#

#=lim_(hto0)((sqrt(3(x+h)-5)-sqrt(3x-5))(sqrt(3(x+h)-5)+sqrt(3x-5)))/(h(sqrt(3(x+h)-5)+sqrt(3x-5))#

#=lim_(hto0)(3(x+h)-5-(3x-5))/(h(sqrt(3(x+h)-5)+sqrt(3x-5))#

#=lim_(hto0)(3x+3h-5-3x+5)/(h(sqrt(3(x+h)-5)+sqrt(3x-5))#

#=lim_(hto0)(3cancel(h))/(cancel(h)sqrt(3(x+h)-5)+sqrt(3x-5))#

#=3/(sqrt(3x-5)+sqrt(3x-5))=3/(2sqrt(3x-5))#

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