Answer to Question #236267 in Calculus for luka

Question #236267

Find the derivative of each of the following functions :

(a)x^2+1 / (2x+1)^1/2

(b)ln(3x^3-9e^x)^3

(c)(x^2+3)√(x)



1
Expert's answer
2021-09-21T12:44:05-0400

(a)


"(\\dfrac{x^2+1}{(2x+1)^{1\/2}})'=((x^2+1)(2x+1)^{-1\/2})'"

"=2x(2x+1)^{-1\/2}-\\dfrac{1}{2}(2)(2x+1)^{-3\/2}(x^2+1)"

"=\\dfrac{4x^2+2x-x^2-1}{(2x+1)^{3\/2}}=\\dfrac{3x^2+2x-1}{(2x+1)^{3\/2}}"

(b)


"(\\ln(3x^3-9e^x)^3)'=3(\\ln(3x^3-9e^x))'"

"=\\dfrac{3(3x^3-9e^x)'}{3x^3-9e^x}=\\dfrac{3(9x^2-9e^x)}{3x^3-9e^x}"

"=\\dfrac{9(x^2-e^x)}{x^3-3e^x}"

(c)


"((x^2+3)\\sqrt{x})'=2x\\sqrt{x}+\\dfrac{x^2+3}{2\\sqrt{x}}"

"=\\dfrac{4x^2+x^2+3}{2\\sqrt{x}}=\\dfrac{5x^2+3}{2\\sqrt{x}}"


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