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For the following equation, evaluate
f^(')(1).

f(x)=2x^(5)+x^(2)-3
Answer:

For the following equation, evaluate $f^{\prime}(1)$ . $\newline$ $f(x)=2 x^{5}+x^{2}-3$ $\newline$ Answer:

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Composition of linear and quadratic functions: find an equation

Full solution

Q. For the following equation, evaluate

f^{\prime}(1)

.

\newline

f(x)=2 x^{5}+x^{2}-3

\newline

Answer:

Apply Power Rule: To find the derivative of the function $f(x) = 2x^5 + x^2 - 3$ , we need to apply the power rule to each term where the power rule states that the derivative of $x^n$ is $n\cdot x^{(n-1)}$ .
Derivative of $2x^5$ : Applying the power rule to the first term: The derivative of $2x^5$ is $5\cdot2x^{5-1} = 10x^4$ .
Derivative of $x^2$ : Applying the power rule to the second term: The derivative of $x^2$ is $2\cdot x^{2-1} = 2x$ .
Derivative of constant: The third term is a constant, $-3$ , and the derivative of a constant is $0$ .
Combine derivatives: Combining the derivatives of all terms, we get the derivative of the function $f(x)$ : $f'(x) = 10x^4 + 2x$ .
Substitute $x=1$ : Now we need to evaluate the derivative at $x = 1$ . Substitute $x$ with $1$ in $f'(x) = 10x^4 + 2x$ .
Evaluate $f'(1)$ : Evaluating the derivative at $x = 1$ : $f'(1) = 10(1)^4 + 2(1) = 10 + 2$ .
Simplify final value: Simplify the expression to find the final value: $f'(1) = 10 + 2 = 12$ .

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