Q. 2.5.2 - 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

a. The derivatives of the three functions are simply constant multiples of f is ${\mathrm{e}}^{2\mathrm{x}},{\mathrm{e}}^{3\mathrm{x}},{\mathrm{e}}^{4\mathrm{x}}$

b. The derivatives of the three functions are algebraic, but the functions are transcendental is $\frac{\mathrm{d}}{\mathrm{dx}}\left({\log }_{\mathrm{b}}\mathrm{x}\right)=\frac{1}{\mathrm{lnbx}},\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{lnx}\right)=\frac{1}{\mathrm{x}},\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\mathrm{x}}$

c. Without the logarithmic differentiation method, finding the three functions are ${\mathrm{x}}^{\mathrm{lnx}},{\left(2\mathrm{x}+1\right)}^{3\mathrm{x}},{\left(\mathrm{lnx}\right)}^{\mathrm{lnx}}$ 

Part (a) Step 1: Given Information

The derivatives of three functions $\mathrm{f}$ are essentially constant multiples of $\mathrm{f}$.

The following concept was used:

The exponential function's features are utilised.

Part (a) Step 2: Calculations

Three functions must be written, with their derivatives being constant multiples of $\mathrm{f}.$

The derivatives of all exponential functions are constant multiplications of the original function.

The given examples are ${\mathrm{e}}^{2\mathrm{x}},{\mathrm{e}}^{3\mathrm{x}},{\mathrm{e}}^{4\mathrm{x}}$

As a result, the derivatives of the three functions are just constant multiples of $\mathrm{f}$ is ${\mathrm{e}}^{2\mathrm{x}},{\mathrm{e}}^{3\mathrm{x}},{\mathrm{e}}^{4\mathrm{x}}$

Part (b) Step 1: Given Information

The derivatives of three functions are algebraic, although the functions themselves are transcendental.

The following concept was used:

The exponential function's features are utilised.

Part (b) Step 2: Calculation

Three functions must be written, with algebraic derivatives but transcendental functions.

The derivatives of logarithmic functions are algebraic, but they are transcendental.

The given examples are $\frac{\mathrm{d}}{\mathrm{dx}}\left({\log }_{\mathrm{b}}\mathrm{x}\right)=\frac{1}{\mathrm{lnbx}},\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{lnx}\right)=\frac{1}{\mathrm{x}},\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\mathrm{x}}$

As a result, the derivatives of the three functions are algebraic, but the functions are transcendental is $\frac{\mathrm{d}}{\mathrm{dx}}\left({\log }_{\mathrm{b}}\mathrm{x}\right)=\frac{1}{\mathrm{lnbx}},\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{lnx}\right)=\frac{1}{\mathrm{x}},\frac{\mathrm{d}}{\mathrm{dx}}\left(\ln \left|\mathrm{x}\right|\right)=\frac{1}{\mathrm{x}}$

Part (c) Step 1: Given Information

Without the logarithmic differentiation method, finding three functions would be difficult or impossible.

The following concept was used:

The exponential function's features are utilised.

Part (c) Step 2: Calculation

It would be difficult or impossible to build three functions without using the logarithmic differentiation method.

The logarithmic differentiation must be used to calculate the derivatives of a function that has variables in both the base and exponent.

The given examples are ${\mathrm{x}}^{\mathrm{lnx}},{\left(2\mathrm{x}+1\right)}^{3\mathrm{x}},{\left(\mathrm{lnx}\right)}^{\mathrm{lnx}}$

Without the approach of logarithmic differentiation, finding the three functions would be difficult or impossible is ${\mathrm{x}}^{\mathrm{lnx}},{\left(2\mathrm{x}+1\right)}^{3\mathrm{x}},{\left(\mathrm{lnx}\right)}^{\mathrm{lnx}}$