How do you find the tangent line approximation to f(x)=1/x near x=1 ?

1 Answer

L(x) = 2-x

We find the answer from the linear approximation L(x) = f(a)+f'(a)(x-a). For linear approximations, we want both f(a) and f'(a) so that they are easy to calculate. In this case, f(1) and f'(1) are easy.

To find the derivative of

f(x) = 1/x

it is best to rewrite it in power form:

f(x) = x^(-1)

Using the power rule for derivatives:

f'(x)=-x^(-2)

So substituting, a=1:

f(a)=1/1

f'(1)=-1/1^2

Substituting this into L(x),

L(x)=1-1(x-1)

=1-x+1

=2-x