If y = tan(3x), find dy/dx.

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Question: If y = tan(3x), find dy/dx.

Options:

Correct Answer: 3sec^2(3x)

Solution:

Using the chain rule, dy/dx = 3sec^2(3x).

If y = tan(3x), find dy/dx.

If y = tan(3x), find dy/dx.

Correct Answer: 3sec^2(3x)

Step 1: Identify the function y = tan(3x).
Step 2: Recognize that this is a composite function where the outer function is tan(u) and the inner function is u = 3x.
Step 3: Recall the derivative of tan(u) is sec^2(u).
Step 4: Apply the chain rule: dy/dx = (dy/du) * (du/dx).
Step 5: Calculate dy/du: since y = tan(u), dy/du = sec^2(u).
Step 6: Substitute u back in: dy/du = sec^2(3x).
Step 7: Calculate du/dx: since u = 3x, du/dx = 3.
Step 8: Combine the results: dy/dx = sec^2(3x) * 3.
Step 9: Simplify the expression: dy/dx = 3sec^2(3x).

Chain Rule – The chain rule is a fundamental differentiation technique used to differentiate composite functions.
Trigonometric Derivatives – Understanding the derivatives of trigonometric functions, particularly the derivative of tangent.