Let
\[
f(x)=-5 x^{2}+5
\]
Find the equation of the tangent line to $f(x)$ at $x=-7$.
\[
y=
\]

Answer

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Answer

$\boxed{y = 70x + 250}$ is the equation of the tangent line to $f(x)$ at $x=-7$

Steps

Step 1 ：Given the function $f(x) = -5x^2 + 5$

Step 2 ：Find the derivative of the function, $f'(x) = -10x$

Step 3 ：Substitute $x = -7$ into the derivative to find the slope of the tangent line, $m = 70$

Step 4 ：Substitute $x = -7$ into the original function to find the y-coordinate of the point of tangency, $y = -240$

Step 5 ：Use the point-slope form of a line to find the equation of the tangent line, $y - (-240) = 70(x - (-7))$

Step 6 ：Simplify to find the equation of the tangent line, $y = 70x + 250$

Step 7 ：$\boxed{y = 70x + 250}$ is the equation of the tangent line to $f(x)$ at $x=-7$