Questions: For the function f(x)=3x^2+9x^3-5x Find the equation of the tangent line at x=-4.

Transcript text: For the function $f(x)=3 x^{2}+9 x^{3}-5 x$ Find the equation of the tangent line at $x=-4$.

Solution

Step 1: Find the derivative of $f(x)$

The derivative of $f(x) = 9 x^{3} + 3 x^{2} - 5 x$ is $f'(x) = 27 x^{2} + 6 x - 5$.

Step 2: Evaluate $f'(a)$

Substituting $x = -4$ into $f'(x)$, we get the slope $m = 403$.

Step 3: Find the y-coordinate of the point

Evaluating $f(-4)$, we find the y-coordinate of the point on the function is $f(-4) = -508$.

Step 4: Use the point-slope form of a line

Using the point-slope form, the equation of the tangent line is $y + 508 = 403(x + 4)$, which simplifies to $y = 403x + 1104$.

The equation of the tangent line to the graph of $f(x) = 9 x^{3} + 3 x^{2} - 5 x$ at $x = -4$ is $y = 403x + 1104$.

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