If y = tan(3x), find dy/dx at x = π/6.
Correct Answer: undefined
- Step 1: Identify the function given in the question, which is y = tan(3x).
- Step 2: To find dy/dx, we need to use the chain rule of differentiation. The derivative of tan(u) is sec^2(u), where u is a function of x.
- Step 3: In our case, u = 3x. Therefore, the derivative dy/dx = d(tan(3x))/dx = sec^2(3x) * d(3x)/dx.
- Step 4: The derivative of 3x with respect to x is 3. So, we have dy/dx = 3 * sec^2(3x).
- Step 5: Now, we need to evaluate dy/dx at x = π/6. Substitute x = π/6 into the derivative: dy/dx = 3 * sec^2(3 * (π/6)).
- Step 6: Simplify the expression: 3 * (π/6) = π/2. So, we need to find sec^2(π/2).
- Step 7: The secant function is the reciprocal of the cosine function. Since cos(π/2) = 0, sec(π/2) = 1/0, which is undefined.
- Step 8: Therefore, sec^2(π/2) is also undefined. This means dy/dx at x = π/6 is undefined.
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