If a function is decreasing, its first derivative is ______________.
If a function is decreasing, its first derivative is negative
If a function is decreasing, its first derivative is negative.
To understand this concept, let’s take a brief review of what a derivative represents. The derivative of a function represents the rate of change of that function at any given point. In other words, it tells us how the function is changing as we move along its graph.
When a function is decreasing, it means that as the input values increase, the output values decrease. Consequently, this implies that the slope of the graph of the function is negative.
The first derivative of a function measures the slope of the tangent line at any given point. If the function is decreasing, then the tangent line at each point will have a negative slope. Therefore, the first derivative of a decreasing function is negative.
Mathematically, if we have a function f(x) and its first derivative is denoted as f'(x), then for a decreasing function:
f'(x) < 0 This tells us that the derivative of a decreasing function is always negative.
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