Calculus Examples

$f (t) = \frac{a (t)}{\sin (x) t \sin (y) t}$

Step 1

Raise $t$ to the power of $1$ .

$\frac{d}{d t} [\frac{a t}{\sin (x) (t^{1} t) \sin (y)}]$

Step 2

Raise $t$ to the power of $1$ .

$\frac{d}{d t} [\frac{a t}{\sin (x) (t^{1} t^{1}) \sin (y)}]$

Step 3

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

$\frac{d}{d t} [\frac{a t}{\sin (x) t^{1 + 1} \sin (y)}]$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Add $1$ and $1$ .

$\frac{d}{d t} [\frac{a t}{\sin (x) t^{2} \sin (y)}]$

Step 4.2

Cancel the common factor of $t$ and $t^{2}$ .

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Step 4.2.1

Factor $t$ out of $a t$ .

$\frac{d}{d t} [\frac{t a}{\sin (x) t^{2} \sin (y)}]$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $t$ out of $\sin (x) t^{2} \sin (y)$ .

$\frac{d}{d t} [\frac{t a}{t (\sin (x) t \sin (y))}]$

Step 4.2.2.2

Cancel the common factor.

$\frac{d}{d t} [\frac{t a}{t (\sin (x) t \sin (y))}]$

Step 4.2.2.3

Rewrite the expression.

$\frac{d}{d t} [\frac{a}{\sin (x) t \sin (y)}]$

Step 5

Since $\frac{a}{\sin (x) \sin (y)}$ is constant with respect to $t$ , the derivative of $\frac{a}{\sin (x) t \sin (y)}$ with respect to $t$ is $\frac{a}{\sin (x) \sin (y)} \frac{d}{d t} [\frac{1}{t}]$ .

$\frac{a}{\sin (x) \sin (y)} \frac{d}{d t} [\frac{1}{t}]$

Step 6

Rewrite $\frac{1}{t}$ as $t^{- 1}$ .

$\frac{a}{\sin (x) \sin (y)} \frac{d}{d t} [t^{- 1}]$

Step 7

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

$\frac{a}{\sin (x) \sin (y)} (- t^{- 2})$

Step 8

Combine fractions.

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Step 8.1

Combine $\frac{a}{\sin (x) \sin (y)}$ and $t^{- 2}$ .

$- \frac{a t^{- 2}}{\sin (x) \sin (y)}$

Step 8.2

Move $t^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{a}{\sin (x) \sin (y) t^{2}}$

Step 9

Simplify.

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Step 9.1

Separate fractions.

$- (\frac{a}{\sin (y) t^{2}} \cdot \frac{1}{\sin (x)})$

Step 9.2

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$- (\frac{a}{\sin (y) t^{2}} \csc (x))$

Step 9.3

Separate fractions.

$- (\frac{a}{t^{2}} \cdot \frac{1}{\sin (y)} \csc (x))$

Step 9.4

Convert from $\frac{1}{\sin (y)}$ to $\csc (y)$ .

$- (\frac{a}{t^{2}} \csc (y) \csc (x))$

Step 9.5

Combine $\frac{a}{t^{2}}$ and $\csc (y)$ .

$- (\frac{a \csc (y)}{t^{2}} \csc (x))$

Step 9.6

Combine $\frac{a \csc (y)}{t^{2}}$ and $\csc (x)$ .

$- \frac{a \csc (y) \csc (x)}{t^{2}}$