Calculus

Differentiation of Functions

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Derivatives of Exponential Functions

Solved Problems

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Example 7

\[y = {2^{ - {x^2}}}\]

Example 8

\[y = {4^x} \cdot {3^{2x}}\]

Example 9

\[y = {x^2}{2^x}.\]

Example 10

\[y = \frac{{{x^2}}}{{{2^x}}}\]

Example 11

\[y = \frac{{{x^n}}}{{{n^x}}}\;\left( {n \gt 0} \right)\]

Example 12

\[y =\frac{{{e^x} - 1}}{{{e^x} + 1}}\]

Example 13

\[y = {e^x}\left( {{x^2} - 2x + 2} \right)\]

Example 14

\[y = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}}\]

Example 15

Show that the function \(y = {e^x}\left( {\sin 2x + \cos 2x} \right)\) is a solution of the differential equation \[y^{\prime\prime} - 2y' + 5y = 0.\]

Example 16

Show that the function \(y = {e^{ - 2x}}\left( {\sin 3x + \cos 3x} \right)\) is a solution of the differential equation \[y^{\prime\prime} + 4y' + 13y = 0.\]

Example 7.

\[y = {2^{ - {x^2}}}\]

Solution.

We find the derivative using the chain rule.

\[y^\prime = \left( {{2^{ - {x^2}}}} \right)^\prime = {2^{ - {x^2}}}\ln 2 \cdot \left( { - {x^2}} \right)^\prime = {2^{ - {x^2}}}\ln 2 \cdot \left( { - 2x} \right) = - x{2^{1 - {x^2}}}\ln 2.\]

Example 8.

\[y = {4^x} \cdot {3^{2x}}\]

Solution.

Using the product and chain rules, we have:

\[y'\left( x \right) = \left( {{4^x} \cdot {3^{2x}}} \right)^\prime = {\left( {{4^x}} \right)^\prime } \cdot {3^{2x}} + {4^x} \cdot {\left( {{3^{2x}}} \right)^\prime } = {4^x}\ln 4 \cdot {3^{2x}} + {4^x} \cdot {3^{2x}}\ln 3 \cdot {\left( {2x} \right)^\prime } = {4^x}\ln 4 \cdot {3^{2x}} + {4^x} \cdot {3^{2x}}\ln 3 \cdot 2 = {4^x} \cdot {3^{2x}}\left( {\ln 4 + 2\ln 3} \right) = {4^x} \cdot {3^{2x}}\left( {\ln 4 + \ln {3^2}} \right) = {4^x} \cdot {3^{2x}}\left( {\ln 4 + \ln 9} \right) = {4^x} \cdot {3^{2x}}\ln 36.\]

There is also an easier way to calculate the derivative. First we simplify the given function by writing it as

\[y = {4^x} \cdot {3^{2x}} = {4^x} \cdot {\left( {{3^2}} \right)^x} = {4^x} \cdot {9^x} = {\left( {4 \cdot 9} \right)^x} = {36^x}.\]

Then the derivative is

\[y'\left( x \right) = {\left( {{4^x} \cdot {3^{2x}}} \right)^\prime } = {\left( {{{36}^x}} \right)^\prime } = {36^x}\ln 36.\]

Example 9.

\[y = {x^2}{2^x}.\]

Solution.

We use the product rule:

\[y^\prime = \left( {{x^2}{2^x}} \right)^\prime = \left( {{x^2}} \right)^\prime \cdot {2^x} + {x^2} \cdot \left( {{2^x}} \right)^\prime = 2x \cdot {2^x} + {x^2} \cdot {2^x}\ln 2 = x{2^x}\left( {2 + x\ln 2} \right).\]

Example 10.

\[y = \frac{{{x^2}}}{{{2^x}}}\]

Solution.

Using the quotient rule, we find:

\[\require{cancel} y'\left( x \right) = \left( {\frac{{{x^2}}}{{{2^x}}}} \right)^\prime = \frac{{{{\left( {{x^2}} \right)}^\prime } \cdot {2^x} - {x^2} \cdot {{\left( {{2^x}} \right)}^\prime }}}{{{{\left( {{2^x}} \right)}^2}}} = \frac{{2 x \cdot {2^x} - {x^2} \cdot {2^x}\ln 2}}{{{{\left( {{2^x}} \right)}^2}}} = \frac{{x\cancel{2^x}\left( {2 - x\ln 2} \right)}}{{{{\left( {{2^x}} \right)}^{\cancel{2}}}}} = \frac{{x\left( {2 - x\ln 2} \right)}}{{{2^x}}}.\;\;\]

Example 11.

\[y = \frac{{{x^n}}}{{{n^x}}}\;\left( {n \gt 0} \right)\]

Solution.

Here the numerator and denominator contain, respectively, a power and an exponential function. Using the quotient rule, we obtain:

\[y'\left( x \right) = \left( {\frac{{{x^n}}}{{{n^x}}}} \right)^\prime = \frac{{{{\left( {{x^n}} \right)}^\prime } \cdot {n^x} - {x^n} \cdot {{\left( {{n^x}} \right)}^\prime }}}{{{{\left( {{n^x}} \right)}^2}}} = \frac{{n {x^{n - 1}} \cdot {n^x} - {x^n} \cdot {n^x}\ln n}}{{{{\left( {{n^x}} \right)}^2}}} = \frac{{{x^{n - 1}}\cancel{n^x}\left( {n - x\ln n} \right)}}{{{{\left( {{n^x}} \right)}^{\cancel{2}}}}} = \frac{{{x^{n - 1}}\left( {n - x\ln n} \right)}}{{{n^x}}}.\]

Example 12.

\[y =\frac{{{e^x} - 1}}{{{e^x} + 1}}\]

Solution.

We use here the quotient rule:

\[y^\prime = \left( {\frac{{{e^x} - 1}}{{{e^x} + 1}}} \right)^\prime = \frac{{{e^x}\left( {{e^x} + 1} \right) - \left( {{e^x} - 1} \right){e^x}}}{{{{\left( {{e^x} + 1} \right)}^2}}} = \frac{{{\cancel{e^{2x}}} + {e^x} - \cancel{e^{2x}} + {e^x}}}{{{{\left( {{e^x} + 1} \right)}^2}}} = \frac{{2{e^x}}}{{{{\left( {{e^x} + 1} \right)}^2}}}.\]

Example 13.

\[y = {e^x}\left( {{x^2} - 2x + 2} \right)\]

Solution.

Using the product rule and the power rule, we have

\[y^\prime = \left[ {{e^x}\left( {{x^2} - 2x + 2} \right)} \right]^\prime = \left( {{e^x}} \right)^\prime \cdot \left( {{x^2} - 2x + 2} \right) + {e^x} \cdot \left( {{x^2} - 2x + 2} \right)^\prime = {e^x}\left( {{x^2} - 2x + 2} \right) + {e^x}\left( {2x - 2} \right) = {e^x}\left( {\color{darkgreen}{x^2} - \cancel{\color{blue}{2x}} + \cancel{\color{red}{2}} + \cancel{\color{blue}{2x}} - \cancel{\color{red}{2}}} \right) = {e^x}\color{darkgreen}{x^2}.\]

Example 14.

\[y = \frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}}\]

Solution.

Combining the quotient rule and the chain rule, we get:

\[y^\prime = \left( {\frac{{{e^x} - {e^{ - x}}}}{{{e^x} + {e^{ - x}}}}} \right)^\prime = \frac{{\left( {{e^x} + {e^{ - x}}} \right)\left( {{e^x} + {e^{ - x}}} \right) - \left( {{e^x} - {e^{ - x}}} \right)\left( {{e^x} - {e^{ - x}}} \right)}}{{{{\left( {{e^x} + {e^{ - x}}} \right)}^2}}} = \frac{{{{\left( {{e^x} + {e^{ - x}}} \right)}^2} - {{\left( {{e^x} - {e^{ - x}}} \right)}^2}}}{{{{\left( {{e^x} + {e^{ - x}}} \right)}^2}}} = \frac{{\cancel{e^{2x}} + 2 + \cancel{e^{ - 2x}} - \cancel{e^{ - 2x}} + 2 - \cancel{e^{ - 2x}}}}{{{{\left( {{e^x} + {e^{ - x}}} \right)}^2}}} = \frac{4}{{{{\left( {{e^x} + {e^{ - x}}} \right)}^2}}}.\]

Example 15.

Show that the function \(y = {e^x}\left( {\sin 2x + \cos 2x} \right)\) is a solution of the differential equation \[y^{\prime\prime} - 2y' + 5y = 0.\]

Solution.

Calculate the derivatives \(y'\) and \(y^{\prime\prime}:\)

\[y' = \left[ {{e^x}\left( {\sin 2x + \cos 2x} \right)} \right]^\prime = {\left( {{e^x}} \right)^\prime }\left( {\sin 2x + \cos 2x} \right) + {e^x}{\left( {\sin 2x + \cos 2x} \right)^\prime } = {e^x}\left( {\color{blue}{\sin 2x} + \color{red}{\cos 2x}} \right) + {e^x}\left( {\color{red}{2\cos 2x} - \color{blue}{2\sin 2x}} \right) = {e^x}\left( {\color{red}{3\cos 2x} - \color{blue}{\sin 2x}} \right);\]
\[y^{\prime\prime} = \left[ {{e^x}\left( {3\cos 2x - \sin 2x} \right)} \right]^\prime = {\left( {{e^x}} \right)^\prime }\left( {3\cos 2x - \sin 2x} \right) + {e^x}{\left( {3\cos 2x - \sin 2x} \right)^\prime } = {e^x}\left( {\color{red}{3\cos 2x} - \color{blue}{\sin 2x}} \right) + {e^x}\left( { - \color{blue}{6\sin 2x} - \color{red}{2\cos 2x}} \right) = {e^x}\left( {\color{red}{\cos 2x} - \color{blue}{7\sin 2x}} \right).\]

Substitute the derivatives and the function itself into the equation:

\[y^{\prime\prime} - 2y' + 5y = 0,\;\; \Rightarrow {e^x}\left( {\color{red}{\cos 2x} - \color{blue}{7\sin 2x}} \right) - 2{e^x}\left( {\color{red}{3\cos 2x} - \color{blue}{\sin 2x}} \right) + 5{e^x}\left( {\color{blue}{\sin 2x} + \color{red}{\cos 2x}} \right) = 0,\;\; \Rightarrow \color{red}{\cos 2x} - \color{blue}{7\sin 2x} - \color{red}{6\cos 2x} + \color{blue}{2\sin 2x} + \color{blue}{5\sin 2x} + \color{red}{5\cos 2x} = \color{black}{0},\;\; \Rightarrow 0 \equiv 0.\]

Hence, this function is a solution of the differential equation.

Example 16.

Show that the function \(y = {e^{ - 2x}}\left( {\sin 3x + \cos 3x} \right)\) is a solution of the differential equation \[y^{\prime\prime} + 4y' + 13y = 0.\]

Solution.

First we determine the first and second order derivatives of the function:

\[y' = \left[ {{e^{ - 2x}}\left( {\sin 3x + \cos 3x} \right)} \right]^\prime = {\left( {{e^{ - 2x}}} \right)^\prime }\left( {\sin 3x + \cos 3x} \right) + {e^{ - 2x}}{\left( {\sin 3x + \cos 3x} \right)^\prime } = - 2{e^{ - 2x}}\left( {\color{blue}{\sin 3x} + \color{red}{\cos 3x}} \right) + {e^{ - 2x}}\left( {\color{red}{3\cos 3x} - \color{blue}{3\sin 3x}} \right) = {e^{ - 2x}}\left( {\color{red}{\cos 3x} - \color{blue}{5\sin 3x}} \right);\]
\[y^{\prime\prime} = \left[ {{e^{ - 2x}}\left( {\cos 3x - 5\sin 3x} \right)} \right]^\prime = {\left( {{e^{ - 2x}}} \right)^\prime }\left( {\cos 3x - 5\sin 3x} \right) + {e^{ - 2x}}{\left( {\cos 3x - 5\sin 3x} \right)^\prime } = - 2{e^{ - 2x}}\left( {\color{red}{\cos 3x} - \color{blue}{5\sin 3x}} \right) + {e^{ - 2x}}\left( { - \color{blue}{3\sin 3x} - \color{red}{15\cos 3x}} \right) = {e^{ - 2x}}\left( {\color{blue}{7\sin 3x} - \color{red}{17\cos 3x}} \right).\]

Substituting the derivatives and the function into the equation yields:

\[y^{\prime\prime} + 4y' + 13y = 0,\;\; \Rightarrow {e^{ - 2x}}\left( {\color{blue}{7\sin 3x} - \color{red}{17\cos 3x}} \right) + 4{e^{ - 2x}}\left( {\color{red}{\cos 3x} - \color{blue}{5\sin 3x}} \right) + 13{e^{ - 2x}}\left( {\color{blue}{\sin 3x} + \color{red}{\cos 3x}} \right) = 0,\;\; \Rightarrow \color{blue}{7\sin 3x} - \color{red}{17\cos 3x} + \color{red}{4\cos 3x} - \color{blue}{20\sin 3x} + \color{blue}{13\sin 3x} + \color{red}{13\cos 3x} = \color{black}{0},\;\; \Rightarrow 0 \equiv 0.\]

So, the function is a solution of the differential equation.

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