Learn how to find the derivative of sqrt(cos x). The derivative is d/dx[sqrt(cos x)].
Below is the graph of f(x) = sqrt(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.
The function sqrt(cos x) is a composition of two functions: the square root function and the cosine function. Since the square root can be rewritten as a power of 1/2, the Power Rule is applicable. Additionally, the cosine function inside the square root requires us to use the chain rule for proper differentiation.
💡 Why this works: We recognize that sqrt(cos x) is a composition of functions, and thus, the Power Rule can be applied to differentiate the square root. The chain rule is also needed to differentiate the cosine function inside the square root.
First, rewrite sqrt(cos x) as (cos x)^(1/2). Then, apply the Power Rule: differentiate the outer function (cos x)^(1/2) and multiply by the derivative of the inner function, cos x. The derivative of (cos x)^(1/2) is (1/2)(cos x)^(-1/2). Then, differentiate cos x, which gives -sin x.
💡 Why this works: The derivative of (cos x)^(1/2) is (1/2)(cos x)^(-1/2). The chain rule requires us to multiply this by the derivative of cos x, which is -sin x. Thus, we arrive at the derivative: d/dx[sqrt(cos x)] = -sin x / (2sqrt(cos x)).
After applying the Power Rule and chain rule, we simplify the expression. The final form of the derivative is -sin x / (2sqrt(cos x)), which is the correct derivative of sqrt(cos x).
💡 Why this works: The simplified derivative, -sin x / (2sqrt(cos x)), is correct because it accurately reflects the rate of change of sqrt(cos x) with respect to x, taking both the square root and the cosine function into account.
[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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