Derivative at a Point
When you are asked for the derivative at a point, you are finding the slope of the tangent line at that specific input value. In practice, this usually means computing the derivative function first, then substituting the given point.
Method
- Differentiate the function using the usual rules: power rule, sum/difference rule, constant rule, product rule, quotient rule, or chain rule as needed.
- Simplify the derivative before plugging in the point. This reduces arithmetic mistakes.
- Evaluate at the given value by replacing the variable with the point from the problem.
- Simplify the final result completely.
What to watch for
- Be careful with negative signs and exponents.
- If the function has several terms, differentiate each term separately.
- If the expression contains products or compositions, apply the correct rule before evaluating.
Quick check
A good check is to see whether your answer is reasonable from the shape or behavior of the function near that point. Also, if possible, substitute the point again into your derivative to confirm the arithmetic. Your final answer should be a single simplified number or expression.