Is the derivative of a function the tangent line?

Question

The question says: The derivative of a function f(x) at x=a is the tangent line to the graph of f(x) at x=a. Find what is wrong with this statement.

 

I'm a little weak on supporting my answer but this is my logic. Is it wrong because the derivative of the function is the slope of a secant line. This is why I am questioning myself: I have heard the slope of a secant line be called the difference quotient or slope between two points but I thought the derivative was the whole function starting with f'(x) not just the slope of a secant line.

 

I'm a little confused on how to find what is wrong and how to support it because I still don't understand derivatives very well. please help me!

Answer 1

The statement should say "The derivative of a function f(x) at x=a is the slope of the tangent line to the graph of f(x) at x=a."

 

 

The difference quotient is for finding the derivative.  This type involves taking the limit.

 

The difference quotient you are referring to is average rate of change.  It takes the slope between two points in the function. This represents the secant line.