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For the function
f(x)=-10x^(2)-10 x-8, find the slope of the tangent line at
x=-10.
Answer:

For the function $f(x)=-10 x^{2}-10 x-8$ , find the slope of the tangent line at $x=-10$ . $\newline$ Answer:

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Linear approximation

Full solution

Q. For the function

f(x)=-10 x^{2}-10 x-8

, find the slope of the tangent line at

x=-10

.

\newline

Answer:

Find Derivative: To find the slope of the tangent line to the function at a given point, we need to find the derivative of the function, which gives us the slope of the tangent line at any point $x$ . The function is $f(x) = -10x^2 - 10x - 8$ . We will use the power rule for differentiation, which states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Apply Power Rule: Differentiate the function with respect to $x$ . The derivative of $-10x^2$ is $-20x$ (using the power rule, with $n=2$ ). The derivative of $-10x$ is $-10$ (the derivative of $x$ is $1$ , so $-10$ times $1$ is $-10$ ). The derivative of a constant, $-10x^2$ $1$ , is $-10x^2$ $2$ . So, the derivative $-10x^2$ $3$ is $-10x^2$ $4$ .
Calculate Derivative: Now we need to find the slope of the tangent line at $x = -10$ . We do this by plugging $x = -10$ into the derivative $f'(x)$ . $f'(-10) = -20(-10) - 10$ .
Find Slope: Calculate the value of $f'(-10)$ . $\newline$ $f'(-10) = -20(-10) - 10 = 200 - 10 = 190$ . $\newline$ So, the slope of the tangent line at $x = -10$ is $190$ .

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