Question

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)

Answer 1

(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)

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

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