Question

For `f(x)=x^5-x^3+x` what is the equation of the tangent line at `x=3`?

Answer 1

`y=379x-918`

Explanation:

The gradient of the tangent line to the curve of f at the point 3 will be equal to the derivative of the function at the point 3.

`thereforef'(x)=5x^4-3x^2+1`

`thereforef'(3)=5*3^4-3*3^2+1=379`

But at 3, the function itself has value

`f(3)=3^5-3^3+3=219`

Therefore the point with co-ordinates `(x,y)=(3,219)` is the point of contact of the tangent with the curve and hence lies on the tangent line so satisfies its equation, so we can substitute it in to yield
`y=mx+c=>219=(379)(3)+c`,
from which we obtain `c=-918`.

Thus the equation of the required tangent line is `y=379x-918`.