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Let $x$ and $y$ be functions of $t$ with $y = \tan x$ . If $\frac{dx}{dt} = 5$ , what is $\frac{dy}{dt}$ when $x = \frac{\pi}{4}$ ? $\newline$ Write an exact, simplified answer.

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Write and solve direct variation equations

Full solution

Q. Let

x

and

y

be functions of

t

with

y = \tan x

. If

\frac{dx}{dt} = 5

, what is

\frac{dy}{dt}

when

x = \frac{\pi}{4}

?

\newline

Write an exact, simplified answer.

Identify Relationship and Differentiate: Identify the relationship and differentiate using the chain rule. $\newline$ Given $y = \tan(x)$ , differentiate both sides with respect to $t$ . $\newline$ $\frac{dy}{dt} = \sec^2(x) \cdot \frac{dx}{dt}$
Differentiate with Respect to t: Substitute the given values. $\newline$ $\frac{dx}{dt} = 5$ and $x = \frac{\pi}{4}$ . $\newline$ $\frac{dy}{dt} = \sec^2\left(\frac{\pi}{4}\right) \cdot 5$
Substitute Given Values: Calculate $\sec^2(\frac{\pi}{4})$ .
$\sec(x) = \frac{1}{\cos(x)}$ , so $\sec(\frac{\pi}{4}) = \frac{1}{\cos(\frac{\pi}{4})} = \frac{1}{(\sqrt{2}/2)} = \sqrt{2}$ .
$\sec^2(\frac{\pi}{4}) = (\sqrt{2})^2 = 2$ .
Calculate $\sec^2(\frac{\pi}{4}):$ Final calculation of $\frac{dy}{dt}$ .
$\frac{dy}{dt} = 2 \times 5 = 10$

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