A line is tangent to the curve y=5x^2+1, in a point x=1. What is the equation of tangent line.

The tangent line in a point is the derivative of a
function in that point.

The equation of the tangent line,
in the point x = 1 is:

y - f(1) =
f'(1)(x-1)

We'll calculate f(1), by substituting x by 1 in
the expression of the function:

f(1) = 5*1^2 +
1

f(1) = 5 + 1

f(1) =
6

To calculate f'(1), first we'll have to differentiate the
given function:

f'(x) = (5x^2 +
1)'

f'(x) = 10x

Now, we'll
substitute x by 1 in the expression of the first
derivative:

f'(1) = 10

Now,
we'll substitute f(1) and f'(1) in the expression of the equation of the tangent
line:

y - f(1) = f'(1)(x-1)

y
- 6 = 10(x - 1)

We'll remove the
brackets:

y - 6 = 10x -
10

We'll add 6 both sides:

y =
10x - 10 + 6

y = 10x -
4

The equation of the tangent line, to the
curve y = 5x^2 + 1, is:

y =
10x - 4