Question

find the tangent line, in standard form, to ( y = f(x) ) at the indicated point.
( y = -4x^{3}-7x + 6 ), at ( x = 3 )
(type your answer in standard form.)

Explanation:

Step1: Find the derivative of the function

The derivative of \(y = -4x^{3}-7x + 6\) using the power rule \((x^{n})^\prime=nx^{n - 1}\) is \(y^\prime=f^\prime(x)=-12x^{2}-7\).

Step2: Find the slope of the tangent line at \(x = 3\)

Substitute \(x = 3\) into the derivative: \(f^\prime(3)=-12\times(3)^{2}-7=-12\times9 - 7=-108-7=-115\).

Step3: Find the \(y\) - coordinate of the point on the curve at \(x = 3\)

Substitute \(x = 3\) into \(y=-4x^{3}-7x + 6\): \(y=-4\times(3)^{3}-7\times3 + 6=-4\times27-21 + 6=-108-21 + 6=-123\). So the point \((x_0,y_0)=(3,-123)\).

Step4: Use the point - slope form \(y - y_0=m(x - x_0)\) and convert to standard form \(Ax+By = C\)

The point - slope form is \(y+123=-115(x - 3)\).
Expand: \(y+123=-115x + 345\).
Rearrange to standard form: \(115x+y=222\).

Answer:

\(115x + y=222\)