Determine derivatives of polynomial, rational, radical, and composite functions and find the equation of the tangent line to the curve y = (x^4 + 3x)^3 at x = -2.

Steps to Solve

1. Identify the Function and the Point We start with the function ( y = (x^4 + 3x^3)^2 ) and the point where ( x = -2 ).

2. Calculate the Derivative Use the chain rule to differentiate: Let ( u = x^4 + 3x^3 ).

Then:

$$ y = u^2 $$

Applying the chain rule, we have:

$$ \frac{dy}{dx} = 2u \cdot \frac{du}{dx} $$

Now, we need to find ( \frac{du}{dx} ):

$$ \frac{du}{dx} = 4x^3 + 9x^2 $$

So,

$$ \frac{dy}{dx} = 2(x^4 + 3x^3)(4x^3 + 9x^2) $$

3. Evaluate the Derivative at ( x = -2 ) Substitute ( x = -2 ) into the derivative:

First, calculate ( u ):

$$ u = (-2)^4 + 3(-2)^3 = 16 - 24 = -8 $$

Next,

$$ \frac{du}{dx} = 4(-2)^3 + 9(-2)^2 = 4(-8) + 36 = -32 + 36 = 4 $$

Now substitute back:

$$ \frac{dy}{dx} = 2(-8)(4) = -64 $$

4. Determine the Point on the Curve Calculate ( y ) when ( x = -2 ):

$$ y = (-2)^4 + 3(-2)^3 = 16 - 24 = -8 $$

The point is ((-2, -8)).

5. Write the Equation of the Tangent Line Using the point-slope form ( y - y_1 = m(x - x_1) ):

Here, ( m = -64 ), ( x_1 = -2 ), and ( y_1 = -8 ):

$$ y - (-8) = -64(x - (-2)) $$

Simplifying:

$$ y + 8 = -64(x + 2) $$

So,

$$ y = -64x - 128 - 8 $$ $$ y = -64x - 136 $$