Calculus Examples

$5^{x} + 2 x$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $5^{x} + 2 x$ with respect to $x$ is $\frac{d}{d x} [5^{x}] + \frac{d}{d x} [2 x]$ .

$\frac{d}{d x} [5^{x}] + \frac{d}{d x} [2 x]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$5^{x} \ln (5) + \frac{d}{d x} [2 x]$

Step 1.3

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 1.3.1

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

$5^{x} \ln (5) + 2 \frac{d}{d x} [x]$

Step 1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$5^{x} \ln (5) + 2 \cdot 1$

Step 1.3.3

Multiply $2$ by $1$ .

$f' (x) = 5^{x} \ln (5) + 2$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $5^{x} \ln (5) + 2$ with respect to $x$ is $\frac{d}{d x} [5^{x} \ln (5)] + \frac{d}{d x} [2]$ .

$\frac{d}{d x} [5^{x} \ln (5)] + \frac{d}{d x} [2]$

Step 2.2

Evaluate $\frac{d}{d x} [5^{x} \ln (5)]$ .

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Step 2.2.1

Since $\ln (5)$ is constant with respect to $x$ , the derivative of $5^{x} \ln (5)$ with respect to $x$ is $\ln (5) \frac{d}{d x} [5^{x}]$ .

$\ln (5) \frac{d}{d x} [5^{x}] + \frac{d}{d x} [2]$

Step 2.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$\ln (5) (5^{x} \ln (5)) + \frac{d}{d x} [2]$

Step 2.2.3

Raise $\ln (5)$ to the power of $1$ .

$5^{x} (\ln^{1} (5) \ln (5)) + \frac{d}{d x} [2]$

Step 2.2.4

Raise $\ln (5)$ to the power of $1$ .

$5^{x} (\ln^{1} (5) \ln^{1} (5)) + \frac{d}{d x} [2]$

Step 2.2.5

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

$5^{x} {\ln (5)}^{1 + 1} + \frac{d}{d x} [2]$

Step 2.2.6

Add $1$ and $1$ .

$5^{x} \ln^{2} (5) + \frac{d}{d x} [2]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$5^{x} \ln^{2} (5) + 0$

Step 2.3.2

Add $5^{x} \ln^{2} (5)$ and $0$ .

$f'' (x) = 5^{x} \ln^{2} (5)$

Step 3

Find the third derivative.

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Step 3.1

Since $\ln^{2} (5)$ is constant with respect to $x$ , the derivative of $5^{x} \ln^{2} (5)$ with respect to $x$ is $\ln^{2} (5) \frac{d}{d x} [5^{x}]$ .

$\ln^{2} (5) \frac{d}{d x} [5^{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $5$ .

$\ln^{2} (5) (5^{x} \ln (5))$

Step 3.3

Multiply $\ln^{2} (5)$ by $\ln (5)$ by adding the exponents.

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Step 3.3.1

Move $\ln (5)$ .

$\ln (5) \ln^{2} (5) \cdot 5^{x}$

Step 3.3.2

Multiply $\ln (5)$ by $\ln^{2} (5)$ .

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Step 3.3.2.1

Raise $\ln (5)$ to the power of $1$ .

$\ln^{1} (5) \ln^{2} (5) \cdot 5^{x}$

Step 3.3.2.2

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

${\ln (5)}^{1 + 2} \cdot 5^{x}$

Step 3.3.3

Add $1$ and $2$ .

$\ln^{3} (5) \cdot 5^{x}$

Step 3.4

Reorder the factors of $\ln^{3} (5) \cdot 5^{x}$ .

$f''' (x) = 5^{x} \ln^{3} (5)$