Learn how to find the derivative of sin ax. The derivative is d/dx[sin ax].
Below is the graph of f(x) = sin 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.
To differentiate sin(ax), we first recognize that it's a composition of the sine function and a linear function ax. Since the derivative of sine involves applying the Power Rule to the inside function, we will differentiate sin(ax) by treating 'a' as a constant factor.
💡 Why this works: The function sin(ax) follows a composition of functions, where the outer function is sine, and the inner function is ax. The Power Rule can be applied to the outer function, and we will differentiate the inner function separately.
Using the Power Rule for differentiation, we first differentiate sin(x) to get cos(x). Then, we apply the chain rule to differentiate the inner function ax. The result is cos(ax), and we multiply by the derivative of ax, which is 'a'.
💡 Why this works: Applying the Power Rule to sin(x) gives cos(x), but because our function is sin(ax), we need to differentiate ax. The derivative of ax is 'a', so we get a * cos(ax) as the derivative of sin(ax).
The final result of applying the rules is a * cos(ax). This is the derivative of sin(ax) after applying the Power Rule and the chain rule.
💡 Why this works: The derivative of sin(ax) simplifies to a * cos(ax), confirming that the correct application of the Power Rule and the chain rule gives this result.
[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.
Calculate any derivative instantly with step-by-step solutions. Perfect for students, teachers, and anyone learning calculus.