Calculus Examples

$\log_{5} (9^{s} - 4)$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d s} [f (g (s))]$ is $f' (g (s)) g' (s)$ where $f (s) = \log_{5} (s)$ and $g (s) = 9^{s} - 4$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $9^{s} - 4$ .

$\frac{d}{d u} [\log_{5} (u)] \frac{d}{d s} [9^{s} - 4]$

Step 1.2

The derivative of $\log_{5} (u)$ with respect to $u$ is $\frac{1}{u \ln (5)}$ .

$\frac{1}{u \ln (5)} \frac{d}{d s} [9^{s} - 4]$

Step 1.3

Replace all occurrences of $u$ with $9^{s} - 4$ .

$\frac{1}{(9^{s} - 4) \ln (5)} \frac{d}{d s} [9^{s} - 4]$

Step 2

By the Sum Rule, the derivative of $9^{s} - 4$ with respect to $s$ is $\frac{d}{d s} [9^{s}] + \frac{d}{d s} [- 4]$ .

$\frac{1}{(9^{s} - 4) \ln (5)} (\frac{d}{d s} [9^{s}] + \frac{d}{d s} [- 4])$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d s} [a^{s}]$ is $a^{s} \ln (a)$ where $a$ = $9$ .

$\frac{1}{(9^{s} - 4) \ln (5)} (9^{s} \ln (9) + \frac{d}{d s} [- 4])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $- 4$ is constant with respect to $s$ , the derivative of $- 4$ with respect to $s$ is $0$ .

$\frac{1}{(9^{s} - 4) \ln (5)} (9^{s} \ln (9) + 0)$

Step 4.2

Simplify terms.

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Step 4.2.1

Add $9^{s} \ln (9)$ and $0$ .

$\frac{1}{(9^{s} - 4) \ln (5)} (9^{s} \ln (9))$

Step 4.2.2

Combine $9^{s}$ and $\frac{1}{(9^{s} - 4) \ln (5)}$ .

$\frac{9^{s}}{(9^{s} - 4) \ln (5)} \ln (9)$

Step 4.2.3

Combine $\frac{9^{s}}{(9^{s} - 4) \ln (5)}$ and $\ln (9)$ .

$\frac{9^{s} \ln (9)}{(9^{s} - 4) \ln (5)}$

Step 4.2.4

Apply the distributive property.

$\frac{9^{s} \ln (9)}{9^{s} \ln (5) - 4 \ln (5)}$