find an equation of the tangent line to the curve at the given point. y = x^4 + 9x^2 - x, (1, 9) y =

Explanation:

Step1: Find the derivative of the function

Using the power - rule $(x^n)'=nx^{n - 1}$, if $y=x^{4}+9x^{2}-x$, then $y' = 4x^{3}+18x - 1$.

Step2: Evaluate the derivative at the given x - value

Substitute $x = 1$ into $y'$. So $y'(1)=4(1)^{3}+18(1)-1=4 + 18-1=21$. The slope $m$ of the tangent line at $x = 1$ is 21.

Step3: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,9)$ and $m = 21$. $y-9=21(x - 1)$.

Step4: Simplify the equation

$y-9=21x-21$. $y=21x-21 + 9$. $y=21x-12$.

Answer:

$y = 21x-12$