To compute the derivative of the function f(x) = 3x³ - 2x² + x - 5, we will apply the rules of differentiation. The derivative of a function gives us the rate at which the function's value changes as its input changes.
The differentiation rules we will use include:
Power Rule: If f(x) = x^n, then f'(x) = n * x^(n-1).
Constant Rule: The derivative of a constant is zero.
Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
Now, let's differentiate each term of the function:
Differentiate 3x³:
Using the power rule, we have:
Derivative: 3 * 3x^(3-1) = 9x²
Differentiate -2x²:
Again, applying the power rule:
Derivative: -2 * 2x^(2-1) = -4x
Differentiate x:
The derivative of x (which is x¹) is simply:
Derivative: 1 * x^(1-1) = 1
Differentiate -5:
Since -5 is a constant:
Derivative: 0
Now, we can combine all the derivatives we calculated:
f'(x) = 9x² - 4x + 1 + 0
Thus, the derivative of the function f(x) is:
f'(x) = 9x² - 4x + 1
To illustrate how this derivative can be used, consider the following:
If we want to find the slope of the tangent line to the curve at a specific point, say x = 2, we can substitute x = 2 into the derivative:
f'(2) = 9(2)² - 4(2) + 1 = 9(4) - 8 + 1 = 36 - 8 + 1 = 29
This means that at the point where x = 2, the slope of the tangent line to the curve is 29.
In conclusion, the derivative of the function f(x) = 3x³ - 2x² + x - 5 is f'(x) = 9x² - 4x + 1. Understanding how to differentiate functions is essential in calculus and is widely applicable in various fields such as physics, engineering, and economics.