Learn how to find the derivative of sin(xy). The derivative is d/dx[sin(xy)].
Below is the graph of f(x) = sin(xy) 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.
sin(xy) involves a product of two variables, so it requires careful application of the Trigonometric Rule and the product rule for differentiation.
💡 Why this works: We recognize that sin(xy) is a composition of the sine function with the product of x and y. Therefore, to differentiate it, we need to apply the product rule along with the derivative of the sine function.
Apply the Trigonometric Rule for sin(u) and the product rule for differentiating x and y.
💡 Why this works: Using the Trigonometric Rule, the derivative of sin(u) is cos(u). Then, we differentiate the product xy using the product rule. This results in a combination of both parts.
Simplify the resulting expression to get the final derivative.
💡 Why this works: After applying the differentiation rules, the final derivative of sin(xy) is a combination of terms involving both x and y, ensuring the product rule was correctly applied and that all parts were differentiated as needed.
[Teaching explanation - to be filled]
[Application to this function - to be filled]
❌ Common Mistake:
Confusing the derivatives of sin(x) and cos(x)
✅ Correct Approach:
Remember: d/dx[sin(x)] = cos(x), d/dx[cos(x)] = -sin(x)
❌ 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.