What is the tangent line to f(x) = (x^3 + 3x – 5)^4 at (1,1)?

Answer 1

To find the tangent line we'll need to know the slope, which is why we'll take the derivative. This function is a great candidate for the chain rule.

f'(x)=4(x³+3x-5)³(3x²+3)

Don't bother distributing, because we just want the slope. Plug in x=1 from our point (1,1)

f'(x)=4((1)³+3(1)-5)³(3(1)²+3)

f'(x)=4(1+3-5)³(3+3)=4(-1)³(6)

f'(x)=-24

Now we know that the tangent line has a slope of -24 and goes through the point (1,1), so using point-slope form we find

y-1=-24(x-1)