Find the slope of the tangent to the curve y  = 3 + 4x^2 +2x^3 at the point where x = a

Question

Does it start something like:

 

((-2x^3 + 4x^2 +3) - (-2a^3 + 4a^2 + 3)) /(x - a)

 

If so, where do I go from there? How would I go about simplifying? I am grateful for any help! 

Answer 1

Slope of tangent line means derivative.  First find the derivative of y with respect to x.  Then evaluate the derivative when x=a.

 

y' = 8x + 6x²

 

 

y' = 8a + 6a²

 

 

Answer 2

Is this for a Calculus problem?  If so, here is how you would start.

 

The limit definition of a derivative (slope of tangent line) is:      lim(x->a) = ((f(x+delta x)-f(x))/(delta x)    but I am assuming you already know this process so we can use the short cut rules.  

Using the power rule, find the derivative (slope of tangent line) by multiplying the coefficient of each variable by the exponent, then lower the exponent by one.  Constants become zero.  

So, your equation would be y'=8x+6x²  

Next, your "x" value is replaced by "a" (x=a) 

That makes it 8a+6a² 

 

I hope that helps!  Feel free to contact me if you would like more explanation, I love calculus!