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For the function
f(x)=x^(2)-11, find the slope of the tangent line at
x=-3.
Answer:

For the function $f(x)=x^{2}-11$ , find the slope of the tangent line at $x=-3$ . $\newline$ Answer:

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Find the slope of a tangent line using limits

Full solution

Q. For the function

f(x)=x^{2}-11

, find the slope of the tangent line at

x=-3

.

\newline

Answer:

Calculate Derivative: To find the slope of the tangent line to the function at a specific point, we need to calculate the derivative of the function, which gives us the slope of the tangent line at any point $x$ . The function given is $f(x) = x^2 - 11$ . 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: Applying the power rule to the function $f(x) = x^2 - 11$ , we differentiate with respect to $x$ to find $f'(x)$ , the derivative of the function. $\newline$ $f'(x) = \frac{d}{dx} (x^2) - \frac{d}{dx} (11)$ $\newline$ $f'(x) = 2\cdot x^{2-1} - 0$ $\newline$ $f'(x) = 2x$
Find Slope at $x = -3$ : Now that we have the derivative $f'(x) = 2x$ , we can find the slope of the tangent line at $x = -3$ by substituting $-3$ into the derivative. $\newline$ $f'(-3) = 2*(-3)$ $\newline$ $f'(-3) = -6$ $\newline$ The slope of the tangent line at $x = -3$ is $-6$ .

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