Q. 1 - Calculus

Question

Differentiation review: Without using the chain rule find the derivative of each of the function f that follows some algebra may be required before differentiating

$$\begin{gathered} a\left)f\right(x)={(3x+1)}^4 \\ b)f\left(x\right)={(\frac{1}{x}+\frac{1}{x^2})}^2 \\ c\left)f\right(x)={(x+1)}^2\sqrt{x} \\ d)f\left(x\right)=\frac{{(x+1)}^2}{\sqrt{x}} \end{gathered}$$

Step-by-Step Solution

Answer

The derivatives can be given as

$$\begin{gathered} a)f\text{'}(x)=324x^3+324x^2+108x+12 \\ b)f\text{'}\left(x\right)=\frac{-2}{x^3}-\frac{6}{x^4}-\frac{4}{x^5} \\ c)f\text{'}(x)=\frac{5}{2}{x}^{\frac{3}{2}}+3\sqrt{x}+\frac{1}{2\sqrt{x}} \\ d)f\text{'}\left(x\right)=\frac{3}{2}\sqrt{x}+\frac{1}{\sqrt{x}}-\frac{2}{x\sqrt{x}} \end{gathered}$$

Step 1: Given information

We are given several functions

Part a )Step 1: Find the derivative of the function

We get,

$$\begin{gathered} f\left(x\right)={(3x+1)}^4 \\ f\left(x\right)={81}^4+108x^3+54x^2+12x+1 \\ On differentiating \\ f\text{'}\left(x\right)=324x^3+324x^2+108x+12 \end{gathered}$$

Which is the required answer

Part b) Step 1: Find the derivative of part b

We have,

$$\begin{gathered} f\left(x\right)={(\frac{1}{x}+\frac{1}{x^2})}^2 \\ f\left(x\right)=\frac{1}{x^2}+\frac{2}{x^3}+\frac{1}{x^4} \\ On differentiating \\ f\text{'}\left(x\right)=-\frac{2}{x^3}-\frac{6}{x^4}-\frac{4}{x^5} \end{gathered}$$

Part c) Step 1: Find the derivative of part c

We have,

$$\begin{gathered} f\left(x\right)={(x+1)}^2\sqrt{x}f\left(x\right)=(x^2+2x+1)\sqrt{x} \\ f\left(x\right)=x^{\frac{5}{2}}+2x^{\frac{3}{2}}+\sqrt{x} \\ On differentiating \\ f\text{'}\left(x\right)=\frac{5}{2}{x}^{\frac{3}{2}}+3\sqrt{x}+\frac{1}{2\sqrt{x}} \end{gathered}$$

Which is required derivative

Part d) Step 1: Find the derivative of part d

We have,

$$\begin{gathered} f\left(x\right)=\frac{{(x+1)}^2}{\sqrt{x}} \\ f\left(x\right)=\frac{x^2+2x+1}{\sqrt{x}} \\ f\left(x\right)=x^{\frac{3}{2}}+2\sqrt{x}+\frac{1}{\sqrt{x}} \\ On differentiating \\ f\text{'}\left(x\right)=\frac{3}{2}\sqrt{x}+\frac{1}{\sqrt{x}}-\frac{2}{x\sqrt{x}} \end{gathered}$$

Which is the required derivative

Step 6: Conclusion

The derivatives can be given as 

$$\begin{gathered} a)f\text{'}(x)=324x^3+324x^2+108x+12 \\ b)f\text{'}\left(x\right)=\frac{-2}{x^3}-\frac{6}{x^4}-\frac{4}{x^5} \\ c)f\text{'}(x)=\frac{5}{2}{x}^{\frac{3}{2}}+3\sqrt{x}+\frac{1}{2\sqrt{x}} \\ d)f\text{'}\left(x\right)=\frac{3}{2}\sqrt{x}+\frac{1}{\sqrt{x}}-\frac{2}{x\sqrt{x}} \end{gathered}$$