Question

if f(x) = 3x^2 - 4, find the equation of the tangent line at x = 3

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=3x^{2}-4$ using the power - rule $(x^n)' = nx^{n - 1}$ is $f'(x)=6x$.

Step2: Find the slope of the tangent line at $x = 3$

Substitute $x = 3$ into $f'(x)$. So $m=f'(3)=6\times3 = 18$.

Step3: Find the y - coordinate of the point of tangency

Substitute $x = 3$ into $f(x)$. So $y=f(3)=3\times3^{2}-4=3\times9 - 4=27 - 4=23$. The point of tangency is $(3,23)$.

Step4: Use the point - slope form of a line

The point - slope form is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(3,23)$ and $m = 18$. So $y-23=18(x - 3)$.

Step5: Simplify the equation

$y-23=18x-54$, then $y=18x- 31$.

Answer:

$y = 18x-31$