Find the derivative function f' for the following functions f.
f(x) =3x²+2x−10; a=1

검증된 단계별 안내

Step 1: Identify the function f(x) = 3x^2 + 2x - 10, which is a polynomial function.

Step 2: Recall the power rule for differentiation, which states that the derivative of x^n is n*x^(n-1).

Step 3: Apply the power rule to each term of the function separately. For the first term, 3x^2, the derivative is 2*3*x^(2-1) = 6x.

Step 4: Apply the power rule to the second term, 2x. The derivative of x is 1, so the derivative of 2x is 2*1 = 2.

Step 5: The derivative of a constant, such as -10, is 0. Combine the derivatives of each term to find f'(x) = 6x + 2.

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Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. The derivative is often denoted as f'(x) and represents the slope of the tangent line to the function's graph at any given point.

Power Rule

The Power Rule is a fundamental technique for finding derivatives of polynomial functions. It states that if f(x) = x^n, where n is a real number, then the derivative f'(x) = n*x^(n-1). This rule simplifies the differentiation process, especially for functions involving powers of x, making it easier to compute derivatives quickly.

Evaluating Derivatives at a Point

To evaluate the derivative at a specific point, such as a = 1, you first find the derivative function f'(x) and then substitute the value of x with the point of interest. This gives the slope of the tangent line to the function at that specific point, providing insight into the function's behavior at that location.

Find the derivative function f' for the following functions f.f(x) =3x²+2x−10; a=1