Question

Calculate f'(π/4) when f(x) = 2sin(x)

Answer 1

The derivative of \
`( f(x) = 2\sin(x) \)` is
`\( f'(x) = 2\cos(x) \)`. Evaluating at
`\(x = (\pi)/(4)\)`, where
`\(\cos((\pi)/(4)) = \sin((\pi)/(4)) = (√(2))/(2)\)`, results in
`\( f'((\pi)/(4)) = √(2) \)`.

Explanation:

The derivative of the given function
`\( f(x) = 2\sin(x) \)` is determined using the chain rule, a fundamental concept in calculus. The chain rule stipulates that the derivative of a composite function is the product of the derivative of the outer function and the derivative of the inner function.

In this case, the outer function is the constant factor 2 multiplied by the sine function, and the inner function is
`\(\sin(x)\)`. The derivative of
`\(\sin(x)\)` is
`\(\cos(x)\)`, and when multiplied by the constant factor 2, the overall derivative becomes
`\( f'(x) = 2\cos(x) \)`.

To find
`\( f'((\pi)/(4)) \)`, we substitute (x) with
`\((\pi)/(4)\)` into the derivative expression. At
`\(x = (\pi)/(4)\)`, both sine and cosine functions evaluate to
`\((√(2))/(2)\)`. Multiplying this by the constant factor 2 gives
`\( f'((\pi)/(4)) = 2 * (√(2))/(2) = √(2) \)`. Therefore, the derivative of
`\( f(x) = 2\sin(x) \)` at
`\(x = (\pi)/(4)\)` is equal to
`\(√(2)\)`.

Answer 2

The derivative of f(x) = 2sin(x) evaluated at x = π/4 is f'(π/4) = √2.

Explanation:

To find the derivative of f(x) = 2sin(x), we use the derivative of the sine function, which is cos(x). Applying this to f(x), the derivative becomes f'(x) = 2cos(x). Now, to find f'(π/4), substitute π/4 into the derivative formula: f'(π/4) = 2cos(π/4).

Using the trigonometric identity cos(π/4) = √2 / 2, we get f'(π/4) = 2 * (√2 / 2) = √2. This result indicates the rate of change of the function 2sin(x) at x = π/4 is represented by the square root of 2.

The derivative represents the slope of the tangent line to the curve at a given point. In this case, f'(π/4) = √2 signifies that at x = π/4, the rate at which the function 2sin(x) is changing is √2. This rate of change is crucial in understanding how the function behaves at that specific point and helps in analyzing various aspects of its behavior around that point.