For the following equation, evaluate f^(')(1). f(x)=-4x^(5)+4x Answer:

Given Function: Given the function f(x) = -4x^5 + 4x, 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.Power Rule Application: 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.Derivative of -4x^5: Differentiating the first term -4x^5 with respect to x, we get -4 * 5 * x^(5-1) = -20x^4.Derivative of 4x: Differentiating the second term 4x with respect to x, we get 4 * 1 * x^(1-1) = 4.Combining Derivatives: Combining the derivatives of both terms, we get f'(x) = -20x^4 + 4.Evaluate at x=1: Now we need to evaluate f'(x) at x = 1. So we substitute x with 1 in the derivative we found: f'(1) = -20(1)^4 + 4.Simplify and Final Result: Simplifying the expression, we get f'(1) = -20(1) + 4 = -20 + 4 = -16.

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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}(1)

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

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Answer:

Given Function: Given the function $f(x) = -4x^5 + 4x$ , 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.
Power Rule Application: 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.
Derivative of $-4x^5$ : Differentiating the first term $-4x^5$ with respect to $x$ , we get $-4 \times 5 \times x^{(5-1)} = -20x^4$ .
Derivative of $4x$ : Differentiating the second term $4x$ with respect to $x$ , we get $4 \times 1 \times x^{(1-1)} = 4$ .
Combining Derivatives: Combining the derivatives of both terms, we get $f'(x) = -20x^4 + 4$ .
Evaluate at $x=1$ : Now we need to evaluate $f'(x)$ at $x = 1$ . So we substitute $x$ with $1$ in the derivative we found: $f'(1) = -20(1)^4 + 4$ .
Simplify and Final Result: Simplifying the expression, we get $f'(1) = -20(1) + 4 = -20 + 4 = -16$ .