Learn how to find the derivative of tan ax. The derivative is d/dx[tan ax].
Below is the graph of f(x) = tan ax and its derivative f'(x). Notice how the original function and derivative relate to each other.
Original Function:
Derivative:
This derivative represents the rate of change of the original function. The calculation follows standard differentiation rules as shown in the step-by-step solution below.
The function tan(ax) is a composition of the tangent function and a linear transformation ax. To differentiate tan(ax), we recognize that we need to apply the chain rule. The Power Rule is used here because the tangent function is a trigonometric function, and its derivative follows the standard rules of differentiation for trigonometric functions.
💡 Why this works: The structure of tan(ax) suggests that the derivative must account for the inner function ax, where the chain rule comes into play alongside the standard derivative of the tangent function.
To differentiate tan(ax), we apply the chain rule. First, differentiate tan(u) where u = ax, using the known result that d/dx[tan(u)] = sec²(u). Then, differentiate the inner function ax, which gives a constant factor 'a'.
💡 Why this works: By applying the chain rule, we get: d/dx[tan(ax)] = sec²(ax) * a. This represents how the derivative of tan(ax) is affected both by the tangent function's shape and the linear factor 'a'.
After applying the chain rule, we arrive at the derivative of tan(ax) as a * sec²(ax). This is the simplified form of the derivative.
💡 Why this works: The expression a * sec²(ax) is correct because it reflects the combination of the Power Rule applied to the tangent function and the chain rule applied to the inner function ax.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Not simplifying the final answer
✅ Correct Approach:
Always simplify your derivative to its most reduced form
Double-check your work by comparing your steps with our solution. Use our calculator to verify each step of the differentiation process.
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