Why does adding a constant to a function not change its derivative?
Conditions
- The function is differentiable.
- The added term is a constant (e.g., 1, , or ).
Reasoning, step by step
- Start with a base function, such as .
- Add a constant to the function, resulting in .
- Apply the differentiation operator to the new function.
- Observe that the derivative of the constant is 0.
- Conclude that .
Example
The video demonstrates this by showing that , , and . The mathematical point is that constant terms vanish under differentiation, so the derivative stays fixed even though the original functions change.
Common misconceptions
- Believing that adding a constant shifts the derivative by the same amount.
- Thinking that the derivative depends on the vertical position of the graph rather than its slope.
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