Calculus Examples

$3^{x} + 2021$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $3^{x} + 2021$ with respect to $x$ is $\frac{d}{d x} [3^{x}] + \frac{d}{d x} [2021]$ .

$\frac{d}{d x} [3^{x}] + \frac{d}{d x} [2021]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$3^{x} \ln (3) + \frac{d}{d x} [2021]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $2021$ is constant with respect to $x$ , the derivative of $2021$ with respect to $x$ is $0$ .

$3^{x} \ln (3) + 0$

Step 1.3.2

Add $3^{x} \ln (3)$ and $0$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

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

Step 2.3

Raise $\ln (3)$ to the power of $1$ .

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

Step 2.4

Raise $\ln (3)$ to the power of $1$ .

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

Step 2.5

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

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

Step 2.6

Add $1$ and $1$ .

$f'' (x) = 3^{x} \ln^{2} (3)$

Step 3

Find the third derivative.

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Step 3.1

Since $\ln^{2} (3)$ is constant with respect to $x$ , the derivative of $3^{x} \ln^{2} (3)$ with respect to $x$ is $\ln^{2} (3) \frac{d}{d x} [3^{x}]$ .

$\ln^{2} (3) \frac{d}{d x} [3^{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $3$ .

$\ln^{2} (3) (3^{x} \ln (3))$

Step 3.3

Multiply $\ln^{2} (3)$ by $\ln (3)$ by adding the exponents.

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Step 3.3.1

Move $\ln (3)$ .

$\ln (3) \ln^{2} (3) \cdot 3^{x}$

Step 3.3.2

Multiply $\ln (3)$ by $\ln^{2} (3)$ .

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Step 3.3.2.1

Raise $\ln (3)$ to the power of $1$ .

$\ln^{1} (3) \ln^{2} (3) \cdot 3^{x}$

Step 3.3.2.2

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

${\ln (3)}^{1 + 2} \cdot 3^{x}$

Step 3.3.3

Add $1$ and $2$ .

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

Step 3.4

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

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