Derivative of 1/sin
Understand the Problem
The question is asking for the derivative of the function 1/sin(x). To find the derivative, we will use the quotient rule or recognize that it can also be expressed as csc(x).
Answer
$-\cos(x) \csc^2(x)$
Answer for screen readers
The derivative of the function $\frac{1}{\sin(x)}$ is $-\cos(x) \csc^2(x)$.
Steps to Solve
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Identify the function The function we want to differentiate is $f(x) = \frac{1}{\sin(x)}$.
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Apply the quotient rule To find the derivative of a quotient, we use the formula: $$ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u'v - uv'}{v^2}, $$ where $u = 1$ and $v = \sin(x)$.
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Calculate derivatives of u and v We calculate the derivatives:
- $u' = 0$ (since the derivative of a constant is zero)
- $v' = \cos(x)$ (the derivative of $\sin(x)$).
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Substitute into the quotient rule formula Plugging our derivatives into the quotient rule gives: $$ f'(x) = \frac{0 \cdot \sin(x) - 1 \cdot \cos(x)}{\sin^2(x)} $$
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Simplify the expression This simplifies to: $$ f'(x) = \frac{-\cos(x)}{\sin^2(x)} $$
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Rewrite using cosecant function Recognizing that $\sin^2(x) = \left(\frac{1}{\csc(x)}\right)^2$, we can express the answer as: $$ f'(x) = -\cos(x) \csc^2(x) $$
The derivative of the function $\frac{1}{\sin(x)}$ is $-\cos(x) \csc^2(x)$.
More Information
The derivative we found can also be expressed in different forms. Notably, since $csc(x)$ is the cosecant function, $-\cos(x) \csc^2(x)$ gives insight into how rapidly the function $\frac{1}{\sin(x)}$ changes with respect to $x$. The cosecant function is the reciprocal of sine, and its derivative can illustrate important properties in calculus.
Tips
- Forgetting to apply the quotient rule correctly can lead to incorrect derivatives.
- Confusing the relationships between sine, cosecant, and their derivatives may cause simplification errors.
- Not simplifying the final result after applying the quotient rule.
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