# 12.6.1. Derivatives of Trigonometric Functions

Given that the function is differentiable at , find the values of and .

• A,
• B,
• C,
• D,

### Example

Given that the function is differentiable at , find the values of and .

### Solution

If the function is differentiable at the point , then the function is continuous at the same point. For this reason, since it is given that the function is differentiable at , the function must also be continuous at . In other words, must be equal to . First, use direct substitution to find as follows:

Now use direct substitution to find .

Since must be equal to , it must be true that , so . Also, for to exist, must be continuous at . For this reason, since it is given that the function is differentiable at , must be continuous at . In other words, must be equal to . Next, find as follows:

Now find .

Since must be equal to , it must be true that . This gives the system of equations . Solve for as follows:

Now substitute for in the equation and solve for .

Next, substitute in the equation for and solve for .

So, and .

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