Question

Sketch a graph of a function whose derivative is always positive.

Short answer

The graph of the function \( f(x) = x \) is a straight line ascending from left to right, indicating that its derivative is always positive.

Step-by-step solution

Understand the Concept of Derivatives

Understand that the derivative of a function at a certain point shows the slope of the tangent line at that point. If the derivative is positive, the function is increasing at that point. If the derivative is always positive, the function is always increasing.

Choose a Function

Choose a function with a derivative that is always positive. For simplicity, a linear function like \( f(x) = x \) can be chosen since its derivative is 1 and is always positive.

Sketch the Graph

Sketch a graph showing the function \( f(x) = x \) which is a straight line ascending from left to right. The line starts from the origin (0,0) and grows as x increases. Make sure to clearly show the positive slope of the line.