Is it correct to deduce the tangent of f(x) = 5x^3 + 5x^2 + x + 1 at x = 0.5 to be y = 9.75x - 1.5?

If yes, could you show the workings? I have this software named Graph v4.4 (

http://www.padowan.dk), which deduced the tangent above, and, according to my knowledge the tangent would be calculated by drawing the dy/dx as follows:

15x^2 + 10x + 1 where, x = 0.5.

Am I mistaken?

Lacking experience with your software and not knowing how you are using it, it is not possible to provide a review of its output. However, since tangent lines are typically linear equations, it seems unlikely that the equation of the tangent in a quadratic. Perhaps you are confusing "derivative of the equation" with "slope of the tangent at one particular point".

To determine the equation of the line which is tangent to a particular point, one finds the derivative, evaluates to find the value of the slope

*at that particular point*, and then creates the linear equation with that slope at the given point, using methodology from one's earlier algebra coursework. Note that, since the derivative

*at that point* is 15/4 + 10/2 + 1 = 39/4 = 9.75, the listed equation seems to be headed in the right direction.