Calculus Examples

$y = \ln ({(t + 1)}^{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = \ln (t)$ and $g (t) = {(t + 1)}^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(t + 1)}^{2}$ .

$\frac{d}{d u_{1}} [\ln (u_{1})] \frac{d}{d t} [{(t + 1)}^{2}]$

Step 1.2

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

$\frac{1}{u_{1}} \frac{d}{d t} [{(t + 1)}^{2}]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(t + 1)}^{2}$ .

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d t} [f (g (t))]$ is $f' (g (t)) g' (t)$ where $f (t) = t^{2}$ and $g (t) = t + 1$ .

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Step 2.1

To apply the Chain Rule, set $u_{2}$ as $t + 1$ .

$\frac{1}{{(t + 1)}^{2}} (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d t} [t + 1])$

Step 2.2

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

$\frac{1}{{(t + 1)}^{2}} (2 u_{2} \frac{d}{d t} [t + 1])$

Step 2.3

Replace all occurrences of $u_{2}$ with $t + 1$ .

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

Step 3

Differentiate.

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Step 3.1

Combine $2$ and $\frac{1}{{(t + 1)}^{2}}$ .

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

$\frac{2}{{(t + 1)}^{2}} ((t + 1) (1 + 0))$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

$\frac{2}{{(t + 1)}^{2}} ((t + 1) \cdot 1)$

Step 3.5.2

Multiply $t + 1$ by $1$ .

$\frac{2}{{(t + 1)}^{2}} (t + 1)$

Step 4

Simplify.

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Step 4.1

Reorder the factors of $\frac{2}{{(t + 1)}^{2}} (t + 1)$ .

$(t + 1) \frac{2}{{(t + 1)}^{2}}$

Step 4.2

Cancel the common factor of $t + 1$ .

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Step 4.2.1

Factor $t + 1$ out of ${(t + 1)}^{2}$ .

$(t + 1) \frac{2}{(t + 1) (t + 1)}$

Step 4.2.2

Cancel the common factor.

$(t + 1) \frac{2}{(t + 1) (t + 1)}$

Step 4.2.3

Rewrite the expression.

$\frac{2}{t + 1}$