Answer :
To find the equation of the tangent line to the graph of the function [tex]f(x) = (4x^3 + 6)^2[/tex] at the point [tex](-1, 4)[/tex], we need to follow these steps:
Differentiate the Function:
First, we need to find the derivative [tex]f'(x)[/tex] of the function [tex]f(x) = (4x^3 + 6)^2[/tex]. We will use the chain rule for differentiation:[tex]u = 4x^3 + 6 \quad \Rightarrow \quad f(x) = u^2[/tex]
Differentiate [tex]u[/tex] with respect to [tex]x[/tex]:
[tex]\frac{du}{dx} = 12x^2[/tex]
Then, apply the chain rule to differentiate [tex]f(x)[/tex]:
[tex]f'(x) = 2u \cdot \frac{du}{dx} = 2(4x^3 + 6) \cdot 12x^2 = 24x^2 (4x^3 + 6)[/tex]
Evaluate the Derivative at [tex]x = -1[/tex]:
Now, substitute [tex]x = -1[/tex] into the derivative to find the slope [tex]m[/tex] of the tangent line:[tex]f'(-1) = 24(-1)^2 (4(-1)^3 + 6) = 24 \cdot 1 \cdot ( -4 + 6) = 24 \cdot 2 = 48[/tex]
Equation of the Tangent Line:
With the slope [tex]m = 48[/tex] and the point [tex](-1, 4)[/tex], use the point-slope form of the line equation:[tex]y - y_1 = m(x - x_1)[/tex]
Substituting [tex]y_1 = 4[/tex], [tex]m = 48[/tex], and [tex]x_1 = -1[/tex]:
[tex]y - 4 = 48(x + 1)[/tex]
Simplify the equation:
[tex]y - 4 = 48x + 48[/tex]
[tex]y = 48x + 52[/tex]
Unfortunately, none of the multiple-choice options matches the equation we derived, suggesting there might be a miscalculation or a typo in the options provided in the question. Since there is no exact match:
Based on our calculations, the equation of the tangent line should be [tex]y = 48x + 52[/tex], but if we need to choose from the provided options, further verification is necessary. In any case, revisit optional calculations or seek clarification to ensure complete accuracy.
