Find the derivative of f(x) = x^3 + 2x^2 - 5x + 1.

Answer
To find the derivative of f(x) = x^3 + 2x^2 - 5x + 1, we can use the power rule of differentiation. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1). Applying this rule to each term of the function, we get:
  • The derivative of x^3 is 3x^2
  • The derivative of 2x^2 is 4x
  • The derivative of -5x is -5
  • The derivative of 1 is 0
Therefore, the derivative of f(x) = x^3 + 2x^2 - 5x + 1 is: f'(x) = 3x^2 + 4x - 5 We can also write this in factored form as: f'(x) = (3x + 4)(x - 5) This means that the slope of the tangent line to the graph of f(x) at any point x is given by the expression 3x^2 + 4x - 5.
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