Calculus Examples

$f (x) = {2.3}^{x}$

Step 1

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2.3$ .

$f' (x) = {2.3}^{x} \ln (2.3)$

Step 2

Find the second derivative.

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Step 2.1

Since $\ln (2.3)$ is constant with respect to $x$ , the derivative of ${2.3}^{x} \ln (2.3)$ with respect to $x$ is $\ln (2.3) \frac{d}{d x} [{2.3}^{x}]$ .

$\ln (2.3) \frac{d}{d x} [{2.3}^{x}]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2.3$ .

$\ln (2.3) ({2.3}^{x} \ln (2.3))$

Step 2.3

Raise $\ln (2.3)$ to the power of $1$ .

${2.3}^{x} (\ln^{1} (2.3) \ln (2.3))$

Step 2.4

Raise $\ln (2.3)$ to the power of $1$ .

${2.3}^{x} (\ln^{1} (2.3) \ln^{1} (2.3))$

Step 2.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${2.3}^{x} {\ln (2.3)}^{1 + 1}$

Step 2.6

Add $1$ and $1$ .

$f'' (x) = {2.3}^{x} \ln^{2} (2.3)$

Step 3

Find the third derivative.

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Step 3.1

Since $\ln^{2} (2.3)$ is constant with respect to $x$ , the derivative of ${2.3}^{x} \ln^{2} (2.3)$ with respect to $x$ is $\ln^{2} (2.3) \frac{d}{d x} [{2.3}^{x}]$ .

$\ln^{2} (2.3) \frac{d}{d x} [{2.3}^{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2.3$ .

$\ln^{2} (2.3) ({2.3}^{x} \ln (2.3))$

Step 3.3

Multiply $\ln^{2} (2.3)$ by $\ln (2.3)$ by adding the exponents.

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Step 3.3.1

Move $\ln (2.3)$ .

$\ln (2.3) \ln^{2} (2.3) \cdot {2.3}^{x}$

Step 3.3.2

Multiply $\ln (2.3)$ by $\ln^{2} (2.3)$ .

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Step 3.3.2.1

Raise $\ln (2.3)$ to the power of $1$ .

$\ln^{1} (2.3) \ln^{2} (2.3) \cdot {2.3}^{x}$

Step 3.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\ln (2.3)}^{1 + 2} \cdot {2.3}^{x}$

Step 3.3.3

Add $1$ and $2$ .

$\ln^{3} (2.3) \cdot {2.3}^{x}$

Step 3.4

Reorder the factors of $\ln^{3} (2.3) \cdot {2.3}^{x}$ .

$f''' (x) = {2.3}^{x} \ln^{3} (2.3)$

Step 4

Find the fourth derivative.

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Step 4.1

Since $\ln^{3} (2.3)$ is constant with respect to $x$ , the derivative of ${2.3}^{x} \ln^{3} (2.3)$ with respect to $x$ is $\ln^{3} (2.3) \frac{d}{d x} [{2.3}^{x}]$ .

$\ln^{3} (2.3) \frac{d}{d x} [{2.3}^{x}]$

Step 4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2.3$ .

$\ln^{3} (2.3) ({2.3}^{x} \ln (2.3))$

Step 4.3

Multiply $\ln^{3} (2.3)$ by $\ln (2.3)$ by adding the exponents.

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Step 4.3.1

Move $\ln (2.3)$ .

$\ln (2.3) \ln^{3} (2.3) \cdot {2.3}^{x}$

Step 4.3.2

Multiply $\ln (2.3)$ by $\ln^{3} (2.3)$ .

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Step 4.3.2.1

Raise $\ln (2.3)$ to the power of $1$ .

$\ln^{1} (2.3) \ln^{3} (2.3) \cdot {2.3}^{x}$

Step 4.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\ln (2.3)}^{1 + 3} \cdot {2.3}^{x}$

Step 4.3.3

Add $1$ and $3$ .

$\ln^{4} (2.3) \cdot {2.3}^{x}$

Step 4.4

Reorder the factors of $\ln^{4} (2.3) \cdot {2.3}^{x}$ .

$f^{4} (x) = {2.3}^{x} \ln^{4} (2.3)$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is ${2.3}^{x} \ln^{4} (2.3)$ .

${2.3}^{x} \ln^{4} (2.3)$