Question

What is the derivative?

Answer 1

Refer to the explanation.

Step-by-step explanation:

Derivatives are a fundamental concept in calculus, which is a branch of mathematics that deals with rates of change and how quantities vary with respect to one another. Specifically, derivatives are used to measure the rate of change of a function with respect to its input variable. In simpler terms, derivatives tell us how a function's output value changes as its input value changes.

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To provide an in-depth explanation of derivatives, let's break it down:

(1) Slope of a Curve:

Consider a simple straight line on a coordinate plane. The slope of this line represents how much the y-coordinate changes for a given change in the x-coordinate. In other words, it tells us the rate of change of y with respect to x. For a line, this slope is constant, and we can find it using:

`\rightarrow \text{slope}(m)=\frac{(\text{change in} \ y)}{(\text{change in} \ x)}=(\Delta y)/(\Delta x)`

(2) Average Rate of Change:

For more complicated curves or functions that are not necessarily straight lines, we can still measure the average rate of change between two points on the curve. The average rate of change of a function f(x) over an interval [a, b] is given by:

`\rightarrow \text{average rate of change} = (f(b)-f(a))/(b-a)`

(3) Instantaneous Rate of Change:

While the average rate of change gives us an overall sense of how a function behaves over an interval, it doesn't tell us what's happening at any particular point on the curve. This is where derivatives come into play. The derivative of a function f(x) at a particular point x=a gives us the instantaneous rate of change of the function at that point. In other words, it tells us the slope of the tangent line to the curve at x=a.

(4) Definition of Derivative:

The derivative of a function f(x) with respect to x is denoted by f'(x) or dy/dx and is defined as the limit of the average rate of change as the interval approaches zero. Mathematically, this is expressed as:

`\rightarrow f'(x) = \lim_(h \to 0) (f(x+h)-f(x))/(h)`

The value of this limit at any specific point x=a gives us the slope of the tangent line to the curve at that point.

(5) Interpretation of Derivatives:

• If the derivative f'(x) is positive at a particular point x=a, it means the function is increasing at that point.

• If the derivative f'(x) is negative at a particular point x=a, it means the function is decreasing at that point.

• If the derivative f'(x) is zero at a particular point x=a, it means the function has a stationary point (maximum, minimum, or an inflection point) at that location.

(6) Higher-Order Derivatives:

The derivative of a function can also be differentiated again to obtain its second derivative, third derivative, and so on. These are known as higher-order derivatives and are denoted by f''(x), f'''(x), and so on. Each derivative provides information about the rate of change of the previous derivative.

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There are different ways to calculate derivatives:

Basic Rules:

For simple functions like polynomials, trigonometric functions, and exponential functions, there are basic rules to find their derivatives. For example:

• Constant Rule:

`\rightarrow (d)/(dx)[c]=0`

• Power Rule:

`\rightarrow (d)/(dx)[x^n]=nx^(n-1)`

• The derivative of sine and cosine functions:

`\rightarrow (d)/(dx)[\sin(x)]=\cos(x)\\\\\rightarrow (d)/(dx)[\cos(x)]=-\sin(x)`

Chain Rule:

The chain rule allows us to find the derivative of composite functions.

`\rightarrow (d)/(dx)[f(g(x))]=f'(g(x)) \cdot g'(x)`

Product Rule:

When we have the product of two functions, say f(x)⋅g(x), the derivative is found using the product rule:

`\rightarrow (d)/(dx)[f(x)g(x)]=f(x)g'(x)+g(x)f'(x)`

Quotient Rule:

When we have the division of two functions, say (f(x))/(g(x)), the derivative is found using the quotient rule:

`\rightarrow (d)/(dx)\Big[(f(x))/(g(x))\Big ]=(g(x)f'(x)-f(x)g'(x))/((g(x))^2)`

Implicit Differentiation:

Some equations may not be explicitly in the form y=f(x), but we can still find their derivatives using implicit differentiation. This method is particularly useful for equations involving x and y that are not easily solvable for y explicitly.

Partial Derivatives:

In multivariable calculus, we deal with functions of multiple variables. The partial derivative of a multivariable function with respect to one of its variables measures how the function changes when only that variable is allowed to vary, keeping the other variables constant.

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In conclusion,

Derivatives find wide application in physics, engineering, economics, and computer science. Assisting in optimization, finding extrema, motion analysis, growth rate determination, and other real-world situations. Studying derivatives is essential for understanding function behavior and properties in calculus.