Find the derivative of each function using the limit definition. a. f(x)=x^(2)+3x-5

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Find the derivative of each function using the limit definition. a.  f(x)=x^(2)+3x-5

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Write Limit Definition: Write down the limit definition of the derivative. The derivative of a function f(x) at a point x is given by the limit as h approaches 0 of the difference quotient: f'(x) = lim_(h -> 0) (f(x+h) - f(x))/h.Apply to Function: Apply the limit definition to the function f(x) = x^2 + 3x - 5. We need to calculate the limit as h approaches 0 of the expression: (f(x+h) - f(x))/h = ((x+h)^2 + 3(x+h) - 5 - (x^2 + 3x - 5))/h.Expand Numerator: Expand the numerator of the difference quotient. Expand (x+h)^2 and 3(x+h) in the numerator to get: (x^2 + 2xh + h^2 + 3x + 3h - 5 - x^2 - 3x + 5)/h.Simplify Numerator: Simplify the numerator by canceling out like terms. The terms x^2, 3x, and -5 cancel out with -x^2, -3x, and +5, respectively, leaving us with: (2xh + h^2 + 3h)/h.Factor Out h: Factor out h from the numerator. Factor h from each term in the numerator to get: h(2x + h + 3)/h.Cancel h: Simplify the difference quotient by canceling h. Cancel h in the numerator and denominator to get: 2x + h + 3.Take Limit: Take the limit as h approaches 0. Now we need to find the limit of 2x + h + 3 as h approaches 0, which is simply 2x + 3 since the term involving h vanishes.Write Final Derivative: Write down the final derivative. The derivative of the function f(x) = x^2 + 3x - 5 is f'(x) = 2x + 3.

Find the derivative of each function using the limit definition. $\newline$ a. $\newline$ $f(x)=x^{2}+3x-5$

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Find the derivative of each function using the limit definition.\newlinea. \newlinef(x)=x2+3x−5f(x)=x^{2}+3x-5f(x)=x2+3x−5

Find the derivative of each function using the limit definition. $\newline$ a. $\newline$ $f(x)=x^{2}+3x-5$