The power rule of differentiation is a fundamental rule in calculus that deals with finding the derivative of a function raised to a power. It is a simple and powerful tool that is used extensively in calculus and other fields of mathematics, science, engineering, and economics. In this answer, we will explain the power rule of differentiation, provide examples, and discuss its applications.

What is the Power Rule of Differentiation?

The power rule of differentiation states that the derivative of a function raised to a power is equal to the product of the power and the derivative of the function. In mathematical notation, the power rule can be expressed as:

d/dx [f(x)^n] = n f(x)^(n-1) f'(x)

where f(x) is a differentiable function, n is a constant exponent, and f'(x) is the derivative of f(x) with respect to x.

The power rule can be used to differentiate any function that is raised to a power, including polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions. The power rule is a consequence of the chain rule of differentiation, which states that the derivative of a composite function is equal to the product of the derivatives of the outer and inner functions.

Examples of the Power Rule of Differentiation

Let us consider some examples of how to apply the power rule of differentiation.

Example 1: Differentiate f(x) = x^2

Using the power rule, we have:

d/dx [x^2] = 2 x^(2-1) = 2x

Therefore, the derivative of f(x) = x^2 is f'(x) = 2x.

Example 2: Differentiate f(x) = x^3 – 2x^2 + 5

Using the power rule, we have:

d/dx [x^3 – 2x^2 + 5] = 3x^2 – 4x

Therefore, the derivative of f(x) = x^3 – 2x^2 + 5 is f'(x) = 3x^2 – 4x.

Example 3: Differentiate f(x) = e^(2x)

Using the chain rule and the power rule, we have:

d/dx [e^(2x)] = e^(2x) * d/dx [2x] = 2 e^(2x)

Therefore, the derivative of f(x) = e^(2x) is f'(x) = 2 e^(2x).

Example 4: Differentiate f(x) = sin(x)^2

Using the chain rule and the power rule, we have:

d/dx [sin(x)^2] = 2 sin(x) * d/dx [sin(x)] = 2 sin(x) cos(x)

Therefore, the derivative of f(x) = sin(x)^2 is f'(x) = 2 sin(x) cos(x).

Applications of the Power Rule of Differentiation

The power rule of differentiation has many applications in calculus and other fields of mathematics, science, engineering, and economics. Some of the common applications of the power rule are:

Finding the slope of a tangent line: The derivative of a function gives the slope of the tangent line to the graph of the function at a given point. The power rule can be used to find the slope of the tangent line to a function raised to a power.

Finding critical points: A critical point of a function is a point where the derivative is zero or undefined. The power rule can be used to find the critical points of a function raised to a power.

Optimization problems: Optimization problems involve finding the maximum or minimum value of a function subject to some constraints. The power rule can be used to find the critical points of the function and determine whether they correspond to a maximum or minimum value.

Differential equations: Differential equations are equations that involve derivatives of a function. The power rule can be used to find the derivative of a function in a differential equation and solve the equation.

Physics: The power rule is used extensively in physics to find the rates of change of physical quantities such as position, velocity, acceleration, force, energy, and power.

Conclusion

The power rule of differentiation is a fundamental rule in calculus that deals with finding the derivative of a function raised to a power. It is a simple and powerful tool that is used extensively in calculus and other fields of mathematics, science, engineering, and economics. The power rule can be used to differentiate any function that is raised to a power, including polynomials, rational functions, exponential functions, logarithmic functions, and trigonometric functions. The power rule has many applications in calculus and other fields, including finding the slope of a tangent line, finding critical points, solving optimization problems, solving differential equations, and solving problems in physics.