In our daily lives, we are frequently curious about how much an alteration in one quantity influences the difference in another closely connected quantity. We refer to this as a pace of change. If you drive a car, for instance, you might want to know about how much the quantity of fuel you use affects how far you’ve travelled. In this case, the rate of change would refer to fuel consumption. A big change in the quantity of fuel in your tank is followed by a smaller change in the distance you have travelled if your car has a high fuel consumption rate.

Similarly, sprinters are curious about the relationship between a change in time and their location, also known as the pace of change as velocity. Although they may not have unique names like fuel usage or velocity, other rates of change still exist and are crucial. An agronomist, for example, might need to know about how changing the amount of fertiliser applied to a specific crop affects the crop’s yield. The impact of a product price shift on consumer demand is a topic of interest for economists.

The goal of differential calculus is to describe how linked quantities change precisely. Here are the three fundamental rules of differentiation in differential calculus.

The power and chain rule

The power and chain rules are two fundamental calculus rules that allow us to differentiate functions.

The power rule notes that if we have a function in the form of f(x) = x^n, where n is any real number, then the derivative of the function is f’ (x) = n*x^(n-1). In other words, when we differentiate a power function, we decrease the power by 1 and multiply it by the original coefficient.

The chain rule is used to differentiate composite functions. If we have a function in the form of f(g(x)), where f and g are functions, then it makes sense why the chain rule says that the function’s derivative is f’ (g(x)) * g’ (x). In other words, we differentiate the outer function with regard to the inner function, after which we multiply by the inner function’s derivative.

These rules are essential in solving problems in calculus, as many functions can be written as a combination of power functions and composite functions.

The product rule

The product rule is a calculus rule used to differentiate the two function’s product.

If we have two functions, f(x) and g(x), then the product rule says that the product’s derivative, h(x) = f(x)*g(x), is given by h’ (x) = f’ (x)*g(x) + f(x)*g’ (x).

In other words, to differentiate the two function’s product, we multiply the first function’s derivative by the second, then add the product of the first function and the derivative of the second function.

The product rule is a valuable tool in calculus, as it allows us to differentiate functions that cannot be written in a simple form.

The quotient rule

The quotient rule is a rule of calculus that is used to differentiate the quotient of two functions.

If we have two functions, f(x) and g(x), it stands to reason that the quotient rule implies that their quotient’s derivative, h(x) = f(x)/g(x), is given by h’ (x) = [f’ (x)*g(x) – f(x)*g’ (x)]/g(x)^2.

In other words, to differentiate the quotient of two functions, we multiply the first function’s derivative by the second, then subtract the product of the first function and the derivative of the second function, and divide the result by the square of the denominator.

The quotient rule is a useful tool in calculus, as it allows us to differentiate functions that involve division, and it is often used in the study of rates of change and optimisation.

Conclusion

Differential calculus is a powerful tool in mathematics and science that is used to study rates of change and slopes of curves. It is essential for modelling and analysing complex systems that are continuously changing, and its applications can be found in a wide range of fields. The fundamental concepts of differential calculus, including limits, derivatives, and differentiation rules, are the foundation of advanced calculus and are essential for further study in mathematics and science.

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