Find the Derivative - d/dq natural log of 1+500q^2

$\ln (1 + 500 q^{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d q} [f (g (q))]$ is $f' (g (q)) g' (q)$ where $f (q) = \ln (q)$ and $g (q) = 1 + 500 q^{2}$ .

To apply the Chain Rule, set $u$ as $1 + 500 q^{2}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d q} [1 + 500 q^{2}]$

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

$\frac{1}{u} \frac{d}{d q} [1 + 500 q^{2}]$

Replace all occurrences of $u$ with $1 + 500 q^{2}$ .

$\frac{1}{1 + 500 q^{2}} \frac{d}{d q} [1 + 500 q^{2}]$

Step 2

Differentiate.

By the Sum Rule, the derivative of $1 + 500 q^{2}$ with respect to $q$ is $\frac{d}{d q} [1] + \frac{d}{d q} [500 q^{2}]$ .

$\frac{1}{1 + 500 q^{2}} (\frac{d}{d q} [1] + \frac{d}{d q} [500 q^{2}])$

Since $1$ is constant with respect to $q$ , the derivative of $1$ with respect to $q$ is $0$ .

$\frac{1}{1 + 500 q^{2}} (0 + \frac{d}{d q} [500 q^{2}])$

Add $0$ and $\frac{d}{d q} [500 q^{2}]$ .

$\frac{1}{1 + 500 q^{2}} \frac{d}{d q} [500 q^{2}]$

Since $500$ is constant with respect to $q$ , the derivative of $500 q^{2}$ with respect to $q$ is $500 \frac{d}{d q} [q^{2}]$ .

$\frac{1}{1 + 500 q^{2}} (500 \frac{d}{d q} [q^{2}])$

Combine $500$ and $\frac{1}{1 + 500 q^{2}}$ .

$\frac{500}{1 + 500 q^{2}} \frac{d}{d q} [q^{2}]$

Differentiate using the Power Rule which states that $\frac{d}{d q} [q^{n}]$ is $n q^{n - 1}$ where $n = 2$ .

$\frac{500}{1 + 500 q^{2}} (2 q)$

Combine fractions.

Combine $2$ and $\frac{500}{1 + 500 q^{2}}$ .

$\frac{2 \cdot 500}{1 + 500 q^{2}} q$

Multiply $2$ by $500$ .

$\frac{1000}{1 + 500 q^{2}} q$

Combine $\frac{1000}{1 + 500 q^{2}}$ and $q$ .

$\frac{1000 q}{1 + 500 q^{2}}$

Reorder terms.

$\frac{1000 q}{500 q^{2} + 1}$

Find the Derivative - d/dq natural log of 1+500q^2

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