(19) Find the equation of the tangent line to the graph ​f(x) = 1/x^4 at x = 3.

Question

The equation for the line tangent to ​f(x) = 1/x^4 at x = 3 is ________ (Answer).

Answer 1

The key to these tangent line questions is to find an equation of the form y = mx + b with m equal to the derivative of f.

So first we take the derivative of f:

df/dx = d/dx (1/x⁴) = d/dx ( x^-4) = -4x^-5 = -4/x⁵

*I find it helpful to rewrite things that look like 1/xⁿ as x^-n before using the power rule*

Now, plugging in x=3 gives us df/dx = -4/3⁵ = -4/243

Now that we have m = -4/243 we use the point slope formula to get our line equation:

y = -4/243(x-3) + f(3)

Rearranging then gives us:

y = -4/243x + 4/81 + 1/81 = -4/243x + 5/81 where we used f(3) = 1/81

So we see the final equation for y = mx + b with m = -4/243 and b = 5/81