Questions: Directions: In this problem we are going to calculate the derivative of tan(x) using the Quotient Rule. For Steps 1-3, leave your answers in terms of sin x and cos x. Step 1: Write tan(x) in terms of sin(x) and cos(x). tan(x)= Step 2: Calculate the derivative of this expression using the Quotient Rule. Once you differentiate, it may be helpful to remember the trig identity: sin^2(x)+cos^2(x)=1. d/dx[tan(x)]= Step 3: Rewrite your expression from step 2 in the following format. d/dx[tan(x)]=1/cos(x) Step 4: Finally, rewrite this expression in terms of sec(x) and tan(x). d/dx[tan(x)]=

Directions: In this problem we are going to calculate the derivative of tan(x) using the Quotient Rule. For Steps 1-3, leave your answers in terms of sin x and cos x. Step 1: Write tan(x) in terms of sin(x) and cos(x). tan(x)= Step 2: Calculate the derivative of this expression using the Quotient Rule. Once you differentiate, it may be helpful to remember the trig identity: sin^2(x)+cos^2(x)=1. d/dx[tan(x)]= Step 3: Rewrite your expression from step 2 in the following format. d/dx[tan(x)]=1/cos(x) Step 4: Finally, rewrite this expression in terms of sec(x) and tan(x). d/dx[tan(x)]=
Transcript text: Directions: In this problem we are going to calculate the derivative of $\tan (x)$ using the Quotient Rule. For Steps $1-3$, leave your answers in terms of $\sin x$ and $\cos x$. Step 1: Write $\tan (x)$ in terms of $\sin (x)$ and $\cos (x)$. \[ \tan (x)= \] $\square$ $\square$ $\square$ Step 2: Calculate the derivative of this expression using the Quotient Rule. Once you differentiate, it may be helpful to remember the trig identity: $\sin ^{2}(x)+\cos ^{2}(x)=1$. $\square$ \[ \frac{d}{d x}[\tan (x)]= \] $\square$ $\square$ Step 3: Rewrite your expression from step 2 in the following format. \[ \frac{d}{d x}[\tan (x)]=\frac{1}{\cos (x)} \] $\square$ $\square$ $\square$ Step 4: Finally, rewrite this expression in terms of $\sec (x)$ and $\tan (x)$. \[ \frac{d}{d x}[\tan (x)]=\square \]
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Solution

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Solution Steps

Step 1: Express tan⁡(x)\tan(x)

We can express tan⁡(x)\tan(x) in terms of sin⁡(x)\sin(x) and cos⁡(x)\cos(x) as follows: tan⁡(x)=sin⁡(x)cos⁡(x) \tan(x) = \frac{\sin(x)}{\cos(x)}

Step 2: Calculate the Derivative

Using the Quotient Rule, the derivative of tan⁡(x)\tan(x) is calculated as: ddx[tan⁡(x)]=sin⁡2(x)cos⁡2(x)+1 \frac{d}{dx}[\tan(x)] = \frac{\sin^2(x)}{\cos^2(x)} + 1

Step 3: Simplify the Derivative

By simplifying the expression, we find: ddx[tan⁡(x)]=cos⁡(x)−2 \frac{d}{dx}[\tan(x)] = \cos(x)^{-2}

Step 4: Rewrite in Terms of sec⁡(x)\sec(x)

Recognizing that sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}, we can rewrite the derivative as: ddx[tan⁡(x)]=sec⁡2(x) \frac{d}{dx}[\tan(x)] = \sec^2(x)

Final Answer

Thus, the derivative of tan⁡(x)\tan(x) is: ddx[tan⁡(x)]=sec⁡2(x) \boxed{\frac{d}{dx}[\tan(x)] = \sec^2(x)}

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