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How do you calculate the derivatives of trigonometric expressions?
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To calculate the derivatives of trigonometric expressions, you'll need to know the derivatives of the basic trigonometric functions and apply the differentiation rules (such as the sum, product, and chain rules) as needed. Here are the derivatives of the six basic trigonometric functions:

1. Sine function:
If $f(x)=\sin (x)$, then $f^{\prime}(x)=\cos (x)$

2. Cosine function:
If $f(x)=\cos (x)$, then $f^{\prime}(x)=-\sin (x)$

3. Tangent function:
If $f(x)=\tan (x)$, then $f^{\prime}(x)=\sec ^2(x)$, where $\sec (x)=\frac{1}{\cos (x)}$

4. Cotangent function:
If $f(x)=\cot (x)$, then $f^{\prime}(x)=-\csc ^2(x)$, where $\csc (x)=\frac{1}{\sin (x)}$

5. Secant function:
If $f(x)=\sec (x)$, then $f^{\prime}(x)=\sec (x) \tan (x)$

6. Cosecant function:
If $f(x)=\csc (x)$, then $f^{\prime}(x)=-\csc (x) \cot (x)$
With these derivatives and the differentiation rules (sum, product, quotient, and chain rules), you can compute the derivatives of trigonometric expressions.

Examples:
1. Find the derivative of $f(x)=\sin (2 x)$.
Here, we apply the chain rule: $f^{\prime}(x)=\cos (2 x) \cdot 2=2 \cos (2 x)$.

2. Find the derivative of $f(x)=\cos (x)+\tan (x)$.
We use the sum rule: $f^{\prime}(x)=-\sin (x)+\sec ^2(x)$

3. Find the derivative of $f(x)=\sin (x) \cos (x)$.
We use the product rule: $f^{\prime}(x)=(\cos (x))(\cos (x))+(\sin (x))(-\sin (x))=$ $\cos ^2(x)-\sin ^2(x)$

4. Find the derivative of $f(x)=\frac{\sin (x)}{\cos (x)}$.
We use the quotient rule: $f^{\prime}(x)=\frac{(\cos (x))(\cos (x))-(\sin (x))(-\sin (x))}{(\cos (x))^2}=\frac{\sin ^2(x)+\cos ^2(x)}{\cos ^2(x)}=$ $\sec ^2(x)$

By applying these derivatives and rules, you can compute the derivatives of more complex trigonometric expressions as needed.

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