Differentiate tan.
Understand the Problem
The question is asking for the derivative of the tangent function, which is a fundamental concept in calculus. To solve this, we will apply the rules of differentiation.
Answer
$$ \frac{d}{dx}\tan(x) = \sec^2(x) $$
Answer for screen readers
The derivative of the tangent function is given by: $$ \frac{d}{dx}\tan(x) = \sec^2(x) $$
Steps to Solve
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Recall the definition of the tangent function The tangent function is defined as the ratio of the sine and cosine functions: $$ \tan(x) = \frac{\sin(x)}{\cos(x)} $$
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Apply the Quotient Rule for differentiation To differentiate a function that is a quotient of two functions, we use the Quotient Rule, which is stated as: $$ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} $$ In this case, let $u = \sin(x)$ and $v = \cos(x)$.
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Find derivatives of the numerator and the denominator First, we differentiate $u$ and $v$:
- The derivative of $u = \sin(x)$ is $u' = \cos(x)$.
- The derivative of $v = \cos(x)$ is $v' = -\sin(x)$.
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Substitute in the Quotient Rule formula Now substitute $u$, $v$, $u'$, and $v'$ into the Quotient Rule: $$ \frac{d}{dx}\tan(x) = \frac{\cos(x) \cdot \cos(x) - \sin(x) \cdot (-\sin(x))}{\cos^2(x)} $$
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Simplify the expression Now simplify the numerator: $$ \cos^2(x) + \sin^2(x) = 1 $$ Thus, we can rewrite the derivative as: $$ \frac{d}{dx}\tan(x) = \frac{1}{\cos^2(x)} $$
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Rewrite using secant function Finally, recognize that $\frac{1}{\cos^2(x)}$ is equal to $\sec^2(x)$: $$ \frac{d}{dx}\tan(x) = \sec^2(x) $$
The derivative of the tangent function is given by: $$ \frac{d}{dx}\tan(x) = \sec^2(x) $$
More Information
The tangent function is a crucial concept in trigonometry, and its derivative, $\sec^2(x)$, plays a significant role in various applications, including physics and engineering. This relationship is often used when solving integrals involving tangent functions.
Tips
- Assuming that the derivative of tangent is simply tangent itself; make sure to use the correct differentiation methods like the Quotient Rule.
- Forgetting to simplify after applying the Quotient Rule, which can lead to complicated or incorrect results.
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