How Does the Derivative of a Cubic Function Result in a Tangent Line?

In summary, the first derivative of a function gives the slope of the tangent line at a given point, not the tangent line itself. This can be seen with functions such as y=x^2 and y=x^3, where the first has a linear derivative and the second has a parabolic derivative. The general formula for a tangent line is y=mx+c, where the derivative of the function gives the value for m.
  • #1
tenman
3
0
I'm have trouble understanding a fundamental question of a derivative. So a derivate gives me a tangent line at any given point on a function.

this makes sense for me for a function y=x^2 because the derivative is y'=2x which is a straight line function.

But what about y=x^3 where the derivative is y'=3x^2 this function is a parabola how can a parabola give me a tangent line?

I'm having trouble understand this very basic concept
 
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  • #2
The first derivative of a function gives you the slope of the tangent line, not the tangent line itself. Thus,for some function $f(x)$, the tangent line (using the point-slope formula) at the point $(x_0,f(x_0))$ is given by:

\(\displaystyle y=f'(x_0)(x-x_0)+f(x_0)\)
 
  • #3
Ahh ok thanks I think I get it now.

So the general formula for a straight line is y=mx+c

are you saying that the derivative of a function gives you the m value needed to plug into the straight line formula?
 
  • #4
tenman said:
Ahh ok thanks I think I get it now.

So the general formula for a straight line is y=mx+c

are you saying that the derivative of a function gives you the m value needed to plug into the straight line formula?

Yes, the derivative of the function evaluated at the $x$-coordinate of the tangent point will give you the slope $m$ of the tangent line. :)
 
  • #5
Thanks heaps for that!
 

What is a derivative?

A derivative is a mathematical concept that represents the instantaneous rate of change of a function at a specific point. It is the slope of the tangent line to the function at that point.

How is a derivative calculated?

The derivative of a function can be calculated using the limit definition of the derivative or through various differentiation rules such as the product rule, quotient rule, and chain rule.

What is the significance of the derivative?

The derivative is significant because it allows us to analyze the behavior of a function at a specific point. It helps us understand the rate of change of a function and its concavity (whether it is increasing or decreasing).

What is a tangent curve?

A tangent curve is a curve that touches a function at a specific point and has the same slope as the derivative of the function at that point. It is used to approximate the behavior of a function near that point.

How are tangent curves used in real life?

Tangent curves are used in many real-life applications, such as in physics to calculate velocities and accelerations, in economics to analyze marginal changes, and in engineering to design curved structures and optimize systems.

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