How do you find the equation of tangent line to the curve #f(x)= x^3+x^2+x+1# at x=4?

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Answer 1 · 2016-08-02T17:41:35

Differentiate using the power rule:

#f'(x) = 3x^2 + 2x + 1 #

Now, find the point, #(x, y)#, for which the function passes through.

#f(4) = 4^3 + 4^2 + 4 + 1 = 64 + 16 + 4 + 1 = 85#

Hence, the function and the tangent pass through the point #(4, 85)#.

Find the slope by plugging in your point, #x = a#, into the derivative and evaluating.

#f'(4) = 3(4)^2 + 2(4) + 1#

#f'(4) = 3(16) + 8 + 1 = 57#

We can finally use point-slope form to determine the equation of the tangent.

#y - y_1 = m(x - x_1)#

#y - 85 = 57(x - 4)#

#y - 85 = 57x - 228#

#y = 57x - 143#

In summary, the equation of the tangent is #y = 57x - 153#.

Hopefully this helps!