Learn how to find the derivative of e^(cos x). The derivative is d/dx[e^(cos x)].
Below is the graph of f(x) = e^(cos x) 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 e^(cos x), first recognize that the function involves an exponential function raised to a trigonometric function, cos x. The Power Rule is applied because of the exponentiation with a function inside.
💡 Why this works: Since the function is an exponential with an inner function (cos x), we need to apply the chain rule, but the Power Rule dictates how we handle the exponential part of the function.
To apply the Power Rule to e^(cos x), treat e as a constant base raised to the power of cos x. First, differentiate the exponential part of the function, which gives e^(cos x). Then, differentiate the inner function, cos x, to get -sin x.
💡 Why this works: By applying the chain rule, the derivative of e^(cos x) becomes e^(cos x) multiplied by the derivative of the inner function, cos x, which is -sin x. Therefore, the derivative is e^(cos x) * (-sin x).
The simplified derivative of e^(cos x) is -e^(cos x) * sin x. This step confirms the derivative by applying the chain rule properly.
💡 Why this works: The final form of the derivative, -e^(cos x) * sin x, is correct because it follows from the Power Rule and the chain rule applied to the inner function cos x.
[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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