Question

Use the appropriate differenatal formula to find© the derivative of the given function6)3(16) 96) = (x²-1) ²(2x+115

Answer 1

`\begin{gathered} a)f^(\prime)(x)=\sqrt{1+x^(2)}+\frac{2x^(2)}{3(1+x^(2))^{(2)/(3)}} \\ \\ b)f^(\prime)(x)=(6x(x^(2)-1)^(2)(2x+1)-2(x^(2)-1)^(3))/((2x+1)^(2)) \end{gathered}`

1) We need to differentiate the following functions:

`\begin{gathered} a)\:f(x)=x\sqrt[3]{1+x^2}\:\:\:\:Use\:the\:product\:rule \\ \\ \\ (d)/(dx)\left(x\right)\sqrt[3]{1+x^2}+(d)/(dx)\left(\sqrt[3]{1+x^2}\right)x \\ \\ \\ 1\cdot \sqrt[3]{1+x^2}+\frac{2x}{3\left(1+x^2\right)^{(2)/(3)}}x \\ \\ \sqrt[3]{1+x^2}+\frac{2x^2}{3\left(x^2+1\right)^{(2)/(3)}} \\ \\ f^(\prime)(x)=\sqrt[3]{1+x^2}+\frac{2x^2}{3\left(1+x^2\right)^{(2)/(3)}} \end{gathered}`

Note that we had to use some properties like the Product Rule, and the Chain Rule.

b) We can start out by applying the Quotient Rule:

`\begin{gathered} g(x)=((x^2-1)^3)/((2x+1)) \\ \\ f^(\prime)(x)=((d)/(dx)\left(\left(x^2-1\right)^3\right)\left(2x+1\right)-(d)/(dx)\left(2x+1\right)\left(x^2-1\right)^3)/(\left(2x+1\right)^2) \\ \\ Differentiating\:each\:part\:of\:that\:quotient: \\ \\ ------- \\ (d)/(dx)\left(\left(x^2-1\right)^3\right)=3\left(x^2-1\right)^2(d)/(dx)\left(x^2-1\right)=6x\left(x^2-1\right)^2 \\ \\ (d)/(dx)\left(x^2-1\right)=(d)/(dx)\left(x^2\right)-(d)/(dx)\left(1\right)=2x \\ \\ (d)/(dx)\left(x^2\right)=2x \\ \\ (d)/(dx)\left(1\right)=0 \\ \\ (d)/(dx)\left(2x+1\right)=2 \\ \\ Writing\:all\:that\:together: \\ \\ f^(\prime)(x)=(6x\left(x^2-1\right)^2\left(2x+1\right)-2\left(x^2-1\right)^3)/(\left(2x+1\right)^2) \\ \end{gathered}`

Thus, these are the answers.