Q. 4 TB - Calculus

Question

Find the derivative and an antiderivative of each of the following functions:

$$\begin{gathered} f\left(x\right)={\pi }^2 \\ f\left(x\right)={\left(\sqrt[11]{x}\right)}^{-2} \\ f\left(x\right)=\frac{1}{5x} \\ f\left(x\right)=se{c}^2(3x+1) \end{gathered}$$

Step-by-Step Solution

Answer

Derivative of $f\left(x\right)={\pi }^2$is $0$and antiderivative is${\mathrm{\pi }}^2\mathrm{x}.$

Derivative of $f\left(x\right)={\left(\sqrt[11]{x}\right)}^{-2}$is $\frac{-2}{11x^{{\displaystyle \frac{15}{11}}}}$and antiderivative is$\frac{11}{9}{x}^{\frac{9}{11}}.$

Derivative of $f\left(x\right)=\frac{1}{5x}$is $\frac{-1}{5x^2}$and antiderivative is$\frac{1}{5}\log x.$

Derivative of $f\left(x\right)=se{c}^2(3x+1)$is $6se{c}^2(3x+1) \tan (3x+1)$and antiderivative is$\frac{1}{3}\tan (3x+1)$

Step 1. Given information.

The given functions are following.

$$\begin{gathered} f\left(x\right)={\pi }^2 \\ f\left(x\right)={\left(\sqrt[11]{x}\right)}^{-2} \\ f\left(x\right)=\frac{1}{5x} \\ f\left(x\right)=se{c}^2(3x+1) \end{gathered}$$

Step 2. derivative and an antiderivative of f ( x ) = π 2 .

Derivative of $f\left(x\right)={\pi }^2.$

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}{\pi }^2 \\ \frac{d}{dx}f\left(x\right)=0 \end{gathered}$$

antiderivative of $f\left(x\right)={\pi }^2$

$$\begin{gathered} \int f\left(x\right) dx=\int {\pi }^2 dx \\ \int f\left(x\right) dx={\pi }^{2}x+C \end{gathered}$$

Step 3. derivative and an antiderivative of f ( x ) = x 11 - 2

Derivative of $f\left(x\right)={\left(\sqrt[11]{x}\right)}^{-2}.$

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}{\left(\sqrt[11]{x}\right)}^{-2} \\ \frac{d}{dx}f\left(x\right)=\frac{-2}{{\left(\sqrt[11]{x}\right)}^3}\frac{d}{dx}\left(\sqrt[11]{x}\right) \\ \frac{d}{dx}f\left(x\right)=\frac{-2}{{\left(\sqrt[11]{x}\right)}^3}\times \frac{1}{11x^{{\displaystyle \frac{10}{11}}}} \\ \frac{d}{dx}f\left(x\right)=\frac{-2}{11x^{{\displaystyle \frac{15}{11}}}} \end{gathered}$$

antiderivative of$f\left(x\right)={\left(\sqrt[11]{x}\right)}^{-2}.$

$$\begin{gathered} \int f\left(x\right) dx=\int {\left(\sqrt[11]{x}\right)}^{-2}dx \\ \int f\left(x\right) dx=\int x^{\frac{-2}{11}}dx \\ \int f\left(x\right) dx=\frac{11}{9}{x}^{\frac{9}{11}}+C \end{gathered}$$

Step 4. derivative and an antiderivative of f ( x ) = 1 5 x

Derivative of $f\left(x\right)=\frac{1}{5x}$

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}\frac{1}{5x} \\ \frac{d}{dx}f\left(x\right)=\frac{1}{5}\frac{d}{dx}{x}^{-1} \\ \frac{d}{dx}f\left(x\right)=\frac{1}{5}\left(-1x^{-2}\right) \\ \frac{d}{dx}f\left(x\right)=\frac{-1}{5x^2} \end{gathered}$$

antiderivative of $f\left(x\right)=\frac{1}{5x}.$

$$\begin{gathered} \int f\left(x\right) dx=\int \frac{1}{5x}dx \\ \int f\left(x\right) dx=\frac{1}{5}\int \frac{1}{x}dx \\ \int f\left(x\right) dx=\frac{1}{5}\log x+C \end{gathered}$$

Step 5. derivative and an antiderivative of f ( x ) = s e c 2 ( 3 x + 1 )

Derivative of $f\left(x\right)=se{c}^2(3x+1)$

$$\begin{gathered} \frac{d}{dx}f\left(x\right)=\frac{d}{dx}se{c}^2(3x+1) \\ =2sec(3x+1)\frac{d}{dx}sec(3x+1) \\ =2sec(3x+1)·sec(3x+1) ·\tan (3x+1)·3 \\ =6se{c}^2(3x+1) \tan (3x+1) \end{gathered}$$

antiderivative of $f\left(x\right)=se{c}^2(3x+1).$

$$\begin{gathered} \int f\left(x\right)dx=\int se{c}^2(3x+1)dx \\ \int f\left(x\right)dx=\frac{1}{3}\tan (3x+1)+C \end{gathered}$$