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Derivative Formula
In mathematical terms, a derivative of a function of a real variable measures the sensitivity to change of a quantity (a function or dependent variable) which is determined by another quantity (the independent variable). A derivative formula is a fundamental tool of calculus. For example, the derivative of the position of a moving object concerning time is the object’s velocity: This measures how quickly the position of the object changes when time advances.
(d/dx).xn = n. xn-1
The derivative can be interpreted as measuring the instantaneous rate of change in an infinitesimal neighbourhood around that point, while integral formulas measure the average rate of change over an entire interval. This interpretation is very useful in particular when examining sums and limits involving derivatives.
In practice, derivatives may be computed by using specific rules and techniques such as:
- Differentiation from first principles
- Differentiation using familiar algebraic rules
- Differentiation using tables and power series expansions
Example of Derivative Formula
Example 1: Given the function f(x)= x^2-4. Find f'(2)
Solution:
Let’s follow the steps of the derivative formula and calculate f'(2).
f'(x)=2x
f'(2)=2*2 = 4
f’(a) = 4
Example 2: The derivative formula is y = x^2; the power rule is basically plugging in 2, so it is (x^2)’ = 2x. The question asks what happens when you add a constant, say 3:
Solution:
(x^2 + 3)’ =
= (x^2)’ + (3)’
= 2x + 0
= 2x.
List of Use of Derivative Formula
- Calculus is all about finding the rate at which one quantity changes concerning another. It has thousands of applications in real life and helps us understand the world better.
- Differentiation is a method to study the behaviour of quantities when they change continuously. It helps in finding equations for tangents and normal curves, rates of change, maxima and minima, intervals of increase or decrease, concavity, and inflexion points in a curve.
- The derivative formula is used for solving geometric problems like area under or above an arc or area between two curves etc., by using the integration concept of calculus.
Rules of Derivative Formula
Derivative formulas are used to help you find the rate of change for a function. They come in handy when you’re trying to understand how fast a function can go from one value to another.
- Constant Rule: The derivative of [math]c[/math] is [math]0[/math], where c is any constant. This means that if the equation has no variable, it’s not going to have a slope.
- Power Rule: The derivative of [math]x^{n}[/math] is [math]nx^{n-1}[/math]. This one is pretty straightforward: if you’re raising x to a power, your derivative will be that power times x raised to one less than the power.
- Sum Rule: The sum of derivatives equals the derivative of the sum. This one means that if there’s more than one term in an equation, you should just find each derivative and add them up. For example, if you had an equation like [math](x+2)^{2}[/math], you’d first find the derivative of x (which is 1), then multiply that by 2 and get 2,
Mathematics classes are meant for providing students with adequate exposure to the concepts of mathematics and making them confident to handle mathematical problems through simple techniques. Visit Cuemath for more information.
