Calculus Examples

$y = 7^{x} + 1102$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $7^{x} + 1102$ with respect to $x$ is $\frac{d}{d x} [7^{x}] + \frac{d}{d x} [1102]$ .

$\frac{d}{d x} [7^{x}] + \frac{d}{d x} [1102]$

Step 1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $7$ .

$7^{x} \ln (7) + \frac{d}{d x} [1102]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

Since $1102$ is constant with respect to $x$ , the derivative of $1102$ with respect to $x$ is $0$ .

$7^{x} \ln (7) + 0$

Step 1.3.2

Add $7^{x} \ln (7)$ and $0$ .

$f' (x) = 7^{x} \ln (7)$

Step 2

Find the second derivative.

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Step 2.1

Since $\ln (7)$ is constant with respect to $x$ , the derivative of $7^{x} \ln (7)$ with respect to $x$ is $\ln (7) \frac{d}{d x} [7^{x}]$ .

$\ln (7) \frac{d}{d x} [7^{x}]$

Step 2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $7$ .

$\ln (7) (7^{x} \ln (7))$

Step 2.3

Raise $\ln (7)$ to the power of $1$ .

$7^{x} (\ln^{1} (7) \ln (7))$

Step 2.4

Raise $\ln (7)$ to the power of $1$ .

$7^{x} (\ln^{1} (7) \ln^{1} (7))$

Step 2.5

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

$7^{x} {\ln (7)}^{1 + 1}$

Step 2.6

Add $1$ and $1$ .

$f'' (x) = 7^{x} \ln^{2} (7)$

Step 3

Find the third derivative.

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Step 3.1

Since $\ln^{2} (7)$ is constant with respect to $x$ , the derivative of $7^{x} \ln^{2} (7)$ with respect to $x$ is $\ln^{2} (7) \frac{d}{d x} [7^{x}]$ .

$\ln^{2} (7) \frac{d}{d x} [7^{x}]$

Step 3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $7$ .

$\ln^{2} (7) (7^{x} \ln (7))$

Step 3.3

Multiply $\ln^{2} (7)$ by $\ln (7)$ by adding the exponents.

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Step 3.3.1

Move $\ln (7)$ .

$\ln (7) \ln^{2} (7) \cdot 7^{x}$

Step 3.3.2

Multiply $\ln (7)$ by $\ln^{2} (7)$ .

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Step 3.3.2.1

Raise $\ln (7)$ to the power of $1$ .

$\ln^{1} (7) \ln^{2} (7) \cdot 7^{x}$

Step 3.3.2.2

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

${\ln (7)}^{1 + 2} \cdot 7^{x}$

Step 3.3.3

Add $1$ and $2$ .

$\ln^{3} (7) \cdot 7^{x}$

Step 3.4

Reorder the factors of $\ln^{3} (7) \cdot 7^{x}$ .

$f''' (x) = 7^{x} \ln^{3} (7)$