Answer to Question #346264 in Calculus for Elam Voyi

Question

Differentiate:

f(x)= (2x⁴+5x+2)

g(x)= 3x²√6x³+5x²+1

h(x)= 4x²/√x+7

Answer 1

a)

$$f'(x)= 2(4x^3)+5(1)+0$$

$$=8x^3+5$$

b)

$$g'(x)=(3x^2)'\sqrt{6x^3+5x^2+1}$$

$$+3x^2(\sqrt{6x^3+5x^2+1})'$$

$$=6x\sqrt{6x^3+5x^2+1}$$

$$+\dfrac{3x^2}{2\sqrt{6x^3+5x^2+1}}(6x^3+5x^2+1)'$$

$$=6x\sqrt{6x^3+5x^2+1}$$

$$+\dfrac{3x^2(18x^2+10x)}{2\sqrt{6x^3+5x^2+1}}$$

$$=\dfrac{3x(12x^3+10x^2+2+9x^3+5x^2)}{\sqrt{6x^3+5x^2+1}}$$

$$=\dfrac{3x(21x^3+15x^2+2)}{\sqrt{6x^3+5x^2+1}}$$

c)

$$h'(x)=\dfrac{(4x^2)'\sqrt{x+7}-4x^2(\sqrt{x+7})'}{x+7}$$

$$=\dfrac{8x\sqrt{x+7}-4x^2(\dfrac{1}{2\sqrt{x+7}})}{x+7}$$

$$=\dfrac{8x^2+56x-2x^2}{(x+7)^{3/2}}$$

$$=\dfrac{6x^2+56x}{(x+7)^{3/2}}$$