Power Rule
The power rule is one of the most basic and frequently used rules in differentiation. It is used to find the derivative of a function that is a power of x. The rule is expressed as follows:
If f(x) = xn, then f'(x) = nx(n-1).
Example 1: Applying the Power Rule
Let’s consider the function f(x) = x3. To find the derivative of this function using the power rule:
f'(x) = 3x(3-1) f'(x) = 3x2
Here, the derivative f'(x) = 3x2 tells us that the slope of the tangent line to the curve of f(x) = x3 at any point x is given by 3x2.
Example 2: Differentiating Higher Powers
Consider g(x) = x5. Applying the power rule:
g'(x) = 5x(5-1) g'(x) = 5x4
For further exploration of differentiation in specific contexts, see Differentiation of Trigonometric Functions and Differentiation of Exponential and Logarithmic Functions.
Key Points to Remember
- The exponent decreases by 1 when differentiated.
- The original exponent becomes the coefficient of the resulting term.
Constant Rule
The constant rule is utilised when we differentiate a constant or a function that is the product of a constant and a variable raised to a power. The rule is defined as follows:
If f(x) = c, where c is a constant, then f'(x) = 0.
If g(x) = c * xn, then g'(x) = c * nx(n-1).
Example 3: Differentiating a Constant
For the function h(x) = 7, since it’s a constant function, its derivative will be:
h'(x) = 0
Example 4: Differentiating a Constant Times a Variable
Consider the function p(x) = 5x4. To find the derivative:
p'(x) = 5 * 4x(4-1) p'(x) = 20x3
Key Points to Remember
- The derivative of a constant is always zero.
- When a variable term is multiplied by a constant, the constant is retained in the derivative.
To advance your understanding of how these rules apply in complex scenarios, consider looking at Techniques of Integration.
Combining the Rules
In practice, you will often need to differentiate functions that require the use of both the power rule and the constant rule. This is particularly true when working with Higher Order Derivatives and Implicit Differentiation, which provide deeper insights into the behaviour of functions.
Example 5: Differentiating a Polynomial
Consider q(x) = 3x3 + 7x2 - 5. To find the derivative:
q'(x) = d(3x3)/dx + d(7x2)/dx - d(5)/dx q'(x) = 3 * 3x(3-1) + 7 * 2x(2-1) - 0 q'(x) = 9x2 + 14x
Key Points to Remember
- Differentiate each term separately and sum their derivatives.
- Apply the relevant rule to each term based on its form.
Practice Questions
Question 1
Find the derivative of r(x) = 4x2 + 9.
Solution
r'(x) = d(4x2)/dx + d(9)/dx r'(x) = 4 * 2x(2-1) + 0 r'(x) = 8x
Question 2
Differentiate the function s(x) = x4 - 6x3 + 2.
Solution
s'(x) = d(x4)/dx - d(6x3)/dx + d(2)/dx s'(x) = 4x(4-1) - 6 * 3x(3-1) + 0 s'(x) = 4x3 - 18x2
Through these examples and practice questions, you've gained insights into the basic differentiation rules, which will be pivotal in understanding more advanced calculus concepts in subsequent sections. Always remember to apply the power and constant rules where applicable and combine them adeptly to differentiate more complex functions.
FAQ
Yes, the power rule can be applied to negative and fractional exponents. If f(x) = xn, where n is a negative or fractional exponent, the power rule states that f'(x) = nx(n-1). For negative exponents, this rule helps us find the rate of change of functions that describe hyperbolas or other types of rational functions. For fractional exponents, which often describe root functions, the power rule allows us to find the rate of change of these non-polynomial functions. The application remains consistent with the rule, and it is crucial in calculus, especially when dealing with functions involving roots and reciprocals.
The derivative of a function at a particular point provides the slope of the tangent line to the graph of the function at that point. Geometrically, if you were to zoom in closely enough at a point on the graph of a function, the graph and the tangent line would become indistinguishable. This means that locally, the function behaves like a linear function with the slope equal to its derivative. Therefore, the derivative f'(x) gives the rate of change of f(x) with respect to x, and it tells us how steeply the function is increasing or decreasing at any particular point x.
The derivative of a constant equals zero because a constant does not change, and the derivative measures the rate of change. In mathematical terms, if f(x) = c, where c is a constant, then f'(x) = 0. In real-world scenarios, this concept implies that if a quantity is constant and does not change with respect to another variable, then its rate of change or derivative is zero. For example, if a car is at rest (constant position), its speed (rate of change of position) is zero. Similarly, if the price of a fixed-rate product remains constant over time, the rate of change of the price with respect to time is zero. This concept helps in understanding and predicting behaviours of various phenomena where constancy is observed.
When differentiating a constant multiplied by a function, the constant rule is applied alongside other relevant differentiation rules. The constant rule states that the derivative of a constant multiplied by a function is equal to the constant multiplied by the derivative of the function. Mathematically, if f(x) = c * g(x), then f'(x) = c * g'(x), where c is a constant and g(x) is a differentiable function. This rule is particularly useful because it allows us to “factor out” constants when differentiating, simplifying the process and making it easier to apply other differentiation rules to the function g(x).
The power rule is so named because it is used to find the derivative of a function that is a power of x. The rule states that if f(x) = xn, then f'(x) = nx(n-1). The power rule is derived using the limit definition of a derivative. By applying the binomial theorem and taking the limit as h approaches 0, we find that the derivative of xn with respect to x is nx(n-1). This rule simplifies the process of finding derivatives for polynomial functions and is fundamental in calculus, as it allows us to easily find the rate of change of power functions.
Practice Questions
The derivative of the function f(x) = 3x3 + 2x with respect to x can be found using the power rule of differentiation. Applying the rule, we get: f'(x) = d(3x3)/dx + d(2x)/dx f'(x) = 3 * 3x(3-1) + 2 f'(x) = 9x2 + 2
The parabola formed by the derivative f'(x) = 9x2 + 2 has a vertex at the minimum point since the coefficient of x^2 is positive. To find the x-coordinate of the vertex, we use the formula: xvertex = -b/(2a) Since there is no x term in f'(x), b = 0 and a = 9. Thus, xvertex = 0 Substituting x_vertex into f'(x) to find the y-coordinate of the vertex: yvertex = f'(0) = 9 * 02 + 2 = 2 Therefore, the coordinates of the vertex of the parabola formed by the derivative are (0, 2).
To find the derivative of the function g(x) = x4 - 6x3 + 2, we apply the power rule to each term: g'(x) = d(x4)/dx - d(6x3)/dx + d(2)/dx g'(x) = 4x(4-1) - 6 * 3x(3-1) + 0 g'(x) = 4x3 - 18x2
To evaluate the derivative at x = 2, we substitute x with 2: g'(2) = 4 * 23 - 18 * 22 g'(2) = 4 * 8 - 18 * 4 g'(2) = 32 - 72 g'(2) = -40
Thus, the derivative of g(x) = x4 - 6x3 + 2 is g'(x) = 4x3 - 18x2 and its value at x = 2 is -40.
