Q. 2 - Calculus

Question

Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) Three functions $f$ whose derivatives are just constant multiples of $f$.

(b) Three functions that are transcendental, but whose derivatives are algebraic. 

(c) A function whose derivative would be difficult or impossible to find without the method of logarithmic differentiation.

Step-by-Step Solution

Answer

Part (a) The three functions its derivatives are just constant multiples of $f is e^{2x},e^{3x},e^{4x}$.

Part (b) The three functions its derivatives are algebraic but the function are transcendental is $\frac{d}{dx}\left({\log }_{b}x\right)=\frac{1}{\ln bx},\frac{d}{dx}(\ln x)=\frac{1}{x},\frac{d}{dx}\left(\ln \left|x\right|\right)=\frac{1}{x}$.

Part (c) The three functions it’s would be difficult or impossible to find without the method of logarithmic differentiation is $x^{\ln x},{(2x+1)}^{3x}, {(\ln x)}^{\ln x}$.

Part (a) Step 1. Given information

Three functions $f$ its derivatives are just constant multiples of $f$.

Part (a) Step 2. Calculation

Need to write three functions its derivatives are just constant multiples of $f$.

 

All exponential functions have the property that their derivatives are constant multiplies of the original function.

Example: $e^{2x},e^{3x},e^{4x}$

Therefore, the three functions its derivatives are just constant multiples of $f$ is $e^{2x},e^{3x},e^{4x}$.

Part (b) Step 1. Given information

Three functions its derivatives are algebraic but the function are transcendental.

Part (b) Step 2. Calculation

Need to write three functions its derivatives are algebraic but the function are transcendental.

 

Logarithmic functions are transcendental but their derivatives are algebraic.

Example: $\frac{d}{dx}\left({\log }_{b}x\right)=\frac{1}{\ln bx},\frac{d}{dx}(\ln x)=\frac{1}{x},\frac{d}{dx}\left(\ln \left|x\right|\right)=\frac{1}{x}$.

Thus, the three functions its derivatives are algebraic, but the function are transcendental is $\frac{d}{dx}\left({\log }_{b}x\right)=\frac{1}{\ln bx},\frac{d}{dx}(\ln x)=\frac{1}{x},\frac{d}{dx}\left(\ln \left|x\right|\right)=\frac{1}{x}$.

Part (c) Step 1. Given information

Three functions it’s would be difficult or impossible to find without the method of logarithmic differentiation.

Part (c) Step 2. Calculation

Need to write three functions it’s would be difficult or impossible to find without the method of logarithmic differentiation.

 

Take the derivatives of a function that involves the variables in both the base and exponent must use the logarithmic differentiation.

 

Example: $x^{\ln x}, {(2x+1)}^{3x}, {(\ln x)}^{\ln x}$

Therefore, the three functions it’s would be difficult or impossible to find without the method of logarithmic differentiation is $x^{\ln x}, {(2x+1)}^{3x}, {(\ln x)}^{\ln x}$.