For the following equation, evaluate f^(')(2). f(x)=-x^(4)+5x Answer:

Identify Function: Given the function f(x) = -x^4 + 5x, we need to find the derivative of the function, which is denoted by f'(x). The derivative of a function gives us the rate at which the function's value changes at any given point.Apply Power Rule: To find f'(x), we will use the power rule for differentiation, which states that the derivative of x^n with respect to x is n*x^(n-1). We will apply this rule to each term in the function separately.Combine Derivatives: The derivative of the first term -x^4 with respect to x is -4x^3 (using the power rule). The derivative of the second term 5x with respect to x is 5 (since the derivative of x is 1 and 5 is a constant).Evaluate at x=2: Combining the derivatives of both terms, we get f'(x) = -4x^3 + 5.Calculate Final Value: Now we need to evaluate f'(x) at x = 2. We substitute x with 2 in the derivative we found: f'(2) = -4(2)^3 + 5.Calculating the value, f'(2) = -4(8) + 5 = -32 + 5 = -27.

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Algebra 2

Evaluate expression when a complex numbers and a variable term is given

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Q. For the following equation, evaluate

f^{\prime}(2)

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f(x)=-x^{4}+5 x

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Answer:

Identify Function: Given the function $f(x) = -x^4 + 5x$ , we need to find the derivative of the function, which is denoted by $f'(x)$ . The derivative of a function gives us the rate at which the function's value changes at any given point.
Apply Power Rule: To find $f'(x)$ , we will use the power rule for differentiation, which states that the derivative of $x^n$ with respect to $x$ is $n*x^{(n-1)}$ . We will apply this rule to each term in the function separately.
Combine Derivatives: The derivative of the first term $-x^4$ with respect to $x$ is $-4x^3$ (using the power rule). The derivative of the second term $5x$ with respect to $x$ is $5$ (since the derivative of $x$ is $1$ and $5$ is a constant).
Evaluate at $x=2$ : Combining the derivatives of both terms, we get $f'(x) = -4x^3 + 5$ .
Calculate Final Value: Now we need to evaluate $f'(x)$ at $x = 2$ . We substitute $x$ with $2$ in the derivative we found: $f'(2) = -4(2)^3 + 5$ .
Calculate Final Value: Now we need to evaluate $f'(x)$ at $x = 2$ . We substitute $x$ with $2$ in the derivative we found: $f'(2) = -4(2)^3 + 5$ . Calculating the value, $f'(2) = -4(8) + 5 = -32 + 5 = -27$ .