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

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)