Calculus Examples

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

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d t} [\frac{f (t)}{g (t)}]$ is $\frac{g (t) \frac{d}{d t} [f (t)] - f (t) \frac{d}{d t} [g (t)]}{{g (t)}^{2}}$ where $f (t) = t \sin (t)$ and $g (t) = 1 + t$ .

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

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d t} [f (t) g (t)]$ is $f (t) \frac{d}{d t} [g (t)] + g (t) \frac{d}{d t} [f (t)]$ where $f (t) = t$ and $g (t) = \sin (t)$ .

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

Step 3

The derivative of $\sin (t)$ with respect to $t$ is $\cos (t)$ .

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

Step 4

Differentiate.

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Step 4.1

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

$\frac{(1 + t) (t \cos (t) + \sin (t) \cdot 1) - t \sin (t) \frac{d}{d t} [1 + t]}{{(1 + t)}^{2}}$

Step 4.2

Multiply $\sin (t)$ by $1$ .

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

Step 4.3

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

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

Step 4.4

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

$\frac{(1 + t) (t \cos (t) + \sin (t)) - t \sin (t) (0 + \frac{d}{d t} [t])}{{(1 + t)}^{2}}$

Step 4.5

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

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

Step 4.6

Add $0$ and $1$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Expand $(1 + t) (t \cos (t) + \sin (t))$ using the FOIL Method.

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Step 5.1.1.1.1

Apply the distributive property.

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

Step 5.1.1.1.2

Apply the distributive property.

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

Step 5.1.1.1.3

Apply the distributive property.

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

Step 5.1.1.2

Simplify each term.

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Step 5.1.1.2.1

Multiply $t \cos (t)$ by $1$ .

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

Step 5.1.1.2.2

Multiply $\sin (t)$ by $1$ .

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

Step 5.1.1.2.3

Multiply $t$ by $t$ by adding the exponents.

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Step 5.1.1.2.3.1

Move $t$ .

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

Step 5.1.1.2.3.2

Multiply $t$ by $t$ .

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

Step 5.1.1.3

Multiply $- 1$ by $1$ .

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

Step 5.1.2

Combine the opposite terms in $t \cos (t) + \sin (t) + t^{2} \cos (t) + t \sin (t) - t \sin (t)$ .

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Step 5.1.2.1

Subtract $t \sin (t)$ from $t \sin (t)$ .

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

Step 5.1.2.2

Add $t \cos (t) + \sin (t) + t^{2} \cos (t)$ and $0$ .

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

Step 5.2

Reorder terms.

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