Find an equation of the tangent line to y= f(x)= at x=1?

Answer 1

The gradient of the tangent at x=1 will be the same as the graidient of the curve (f(x)) when x = 1

The gradient of the curve is found by differentiating.

d/dx f(x) = f'(x)

When x = 1, f'(x) = f'(1)

This is the gradient of the tangent when x = 1

The general equation of a tangent is y - y1 = m(x - x1)
where m is the gradient and x1 and y1 are co-ordinates which you know the tangent passes through.

we know m is f'(1)

We know x = 1 is the x1 coordinate of a point on the tangent.

We just need the y coordinate.

When x = 1, y = f(1)

Therefore the equation of the tangent is

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

or

y = xf'(1)-f'(1) +f(1)

Answer 2

y = f'(1) + b