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For the function
f(x)=-6x^(2)+9x+2, find the slope of the tangent line at
x=2.
Answer:

For the function $f(x)=-6 x^{2}+9 x+2$ , find the slope of the tangent line at $x=2$ . $\newline$ Answer:

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Find the slope of a tangent line using limits

Full solution

Q. For the function

f(x)=-6 x^{2}+9 x+2

, find the slope of the tangent line at

x=2

.

\newline

Answer:

Calculate Derivative: To find the slope of the tangent line at a specific point on the graph of a function, we need to calculate the derivative of the function at that point. The derivative of a function at a point gives us the slope of the tangent line to the function at that point.
Apply Power Rule: The function given is $f(x) = -6x^2 + 9x + 2$ . We will find the derivative of this function, $f'(x)$ , using the power rule. The power rule states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Evaluate Derivative: Applying the power rule to each term of the function: $\newline$ The derivative of $-6x^2$ is $-12x$ (since the exponent is $2$ , we multiply by $2$ and subtract $1$ from the exponent). $\newline$ The derivative of $9x$ is $9$ (since the exponent is $1$ , the derivative is just the coefficient). $\newline$ The derivative of the constant $2$ is $0$ (since the derivative of any constant is $0$ ). $\newline$ So, $-12x$ $1$ .
Find Slope: Now we need to evaluate the derivative at $x=2$ to find the slope of the tangent line at that point. $f'(2) = -12(2) + 9 = -24 + 9 = -15.$
Find Slope: Now we need to evaluate the derivative at $x=2$ to find the slope of the tangent line at that point. $f'(2) = -12(2) + 9 = -24 + 9 = -15.$ The slope of the tangent line to the function at $x=2$ is $-15$ .

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