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For the function
f(x)=3x^(2)+7x+4, find the slope of the tangent line at
x=6.
Answer:

For the function $f(x)=3 x^{2}+7 x+4$ , find the slope of the tangent line at $x=6$ . $\newline$ Answer:

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Find the slope of a tangent line using limits

Full solution

Q. For the function

f(x)=3 x^{2}+7 x+4

, find the slope of the tangent line at

x=6

.

\newline

Answer:

Calculate Derivative of $f(x)$ : To find the slope of the tangent line to the function $f(x)$ at a specific point $x=6$ , we need to calculate the derivative of $f(x)$ with respect to $x$ , which will give us the slope of the tangent line at any point $x$ . The function is $f(x) = 3x^2 + 7x + 4$ . We will use the power rule for differentiation, which states that the derivative of $x^n$ with respect to $x$ is $n\cdot x^{(n-1)}$ .
Apply Power Rule for Differentiation: First, we differentiate each term of the function separately. $\newline$ The derivative of the first term, $3x^2$ , with respect to $x$ is $2\cdot 3x^{(2-1)} = 6x$ . $\newline$ The derivative of the second term, $7x$ , with respect to $x$ is $7$ . $\newline$ The derivative of the constant term, $4$ , with respect to $x$ is $0$ , since the derivative of a constant is always $0$ .
Combine Derivatives of Terms: Now, we combine the derivatives of all terms to get the derivative of the entire function $f(x)$ . The derivative $f'(x) = 6x + 7$ .
Evaluate Derivative at $x=6$ : Next, we evaluate the derivative at $x=6$ to find the slope of the tangent line at that point. $f'(6) = 6\times6 + 7 = 36 + 7 = 43.$
Find Slope at $x=6$ : The slope of the tangent line to the function $f(x)$ at $x=6$ is $43$ .

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