Calculus Examples

$g (θ) = e^{θ} (\tan (θ) - θ)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d θ} [f (θ) g (θ)]$ is $f (θ) \frac{d}{d θ} [g (θ)] + g (θ) \frac{d}{d θ} [f (θ)]$ where $f (θ) = e^{θ}$ and $g (θ) = \tan (θ) - θ$ .

$e^{θ} \frac{d}{d θ} [\tan (θ) - θ] + (\tan (θ) - θ) \frac{d}{d θ} [e^{θ}]$

Step 2

By the Sum Rule, the derivative of $\tan (θ) - θ$ with respect to $θ$ is $\frac{d}{d θ} [\tan (θ)] + \frac{d}{d θ} [- θ]$ .

$e^{θ} (\frac{d}{d θ} [\tan (θ)] + \frac{d}{d θ} [- θ]) + (\tan (θ) - θ) \frac{d}{d θ} [e^{θ}]$

Step 3

The derivative of $\tan (θ)$ with respect to $θ$ is $\sec^{2} (θ)$ .

$e^{θ} (\sec^{2} (θ) + \frac{d}{d θ} [- θ]) + (\tan (θ) - θ) \frac{d}{d θ} [e^{θ}]$

Step 4

Differentiate.

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Step 4.1

Since $- 1$ is constant with respect to $θ$ , the derivative of $- θ$ with respect to $θ$ is $- \frac{d}{d θ} [θ]$ .

$e^{θ} (\sec^{2} (θ) - \frac{d}{d θ} [θ]) + (\tan (θ) - θ) \frac{d}{d θ} [e^{θ}]$

Step 4.2

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

$e^{θ} (\sec^{2} (θ) - 1 \cdot 1) + (\tan (θ) - θ) \frac{d}{d θ} [e^{θ}]$

Step 4.3

Multiply $- 1$ by $1$ .

$e^{θ} (\sec^{2} (θ) - 1) + (\tan (θ) - θ) \frac{d}{d θ} [e^{θ}]$

Step 5

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

$e^{θ} (\sec^{2} (θ) - 1) + (\tan (θ) - θ) e^{θ}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$e^{θ} \sec^{2} (θ) + e^{θ} \cdot - 1 + (\tan (θ) - θ) e^{θ}$

Step 6.2

Apply the distributive property.

$e^{θ} \sec^{2} (θ) + e^{θ} \cdot - 1 + \tan (θ) e^{θ} - θ e^{θ}$

Step 6.3

Combine terms.

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Step 6.3.1

Move $- 1$ to the left of $e^{θ}$ .

$e^{θ} \sec^{2} (θ) - 1 \cdot e^{θ} + \tan (θ) e^{θ} - θ e^{θ}$

Step 6.3.2

Rewrite $- 1 e^{θ}$ as $- e^{θ}$ .

$e^{θ} \sec^{2} (θ) - e^{θ} + \tan (θ) e^{θ} - θ e^{θ}$

Step 6.4

Reorder terms.

$e^{θ} \sec^{2} (θ) + e^{θ} \tan (θ) - e^{θ} θ - e^{θ}$

Step 6.5

Move $- e^{θ}$ .

$e^{θ} \sec^{2} (θ) - e^{θ} + e^{θ} \tan (θ) - e^{θ} θ$

Step 6.6

Factor $e^{θ}$ out of $e^{θ} \sec^{2} (θ)$ .

$e^{θ} (\sec^{2} (θ)) - e^{θ} + e^{θ} \tan (θ) - e^{θ} θ$

Step 6.7

Factor $e^{θ}$ out of $- e^{θ}$ .

$e^{θ} (\sec^{2} (θ)) + e^{θ} \cdot - 1 + e^{θ} \tan (θ) - e^{θ} θ$

Step 6.8

Factor $e^{θ}$ out of $e^{θ} (\sec^{2} (θ)) + e^{θ} \cdot - 1$ .

$e^{θ} (\sec^{2} (θ) - 1) + e^{θ} \tan (θ) - e^{θ} θ$

Step 6.9

Apply pythagorean identity.

$e^{θ} \tan^{2} (θ) + e^{θ} \tan (θ) - e^{θ} θ$

Step 6.10

Reorder factors in $e^{θ} \tan^{2} (θ) + e^{θ} \tan (θ) - e^{θ} θ$ .

$e^{θ} \tan^{2} (θ) + e^{θ} \tan (θ) - θ e^{θ}$