Differential can seem like a lengthy and complicated process when done through first principles. Luckily for math students everywhere, there are a number of rules which can be used to preform differentiation much more quickly. Here I will outline four rules commonly taught in high school calculus courses: the power rule, the product rule, the quotient rule, and the chain rule.
The Power Rule
The power rule is one of the most important differentiation rules in modern calculus. It can be used to differentiate polynomials since differentiation is linear. This is done by multiplying the variable by the value of it’s exponent, n, and then subtracting one from the original exponent, as shown below.
d(x^n)/dx = nx^(n-1)
Example: f(x) = x^3
d(f(x))/dx = 3x^2
Example:g(x) = 2x^5
d(g(x))/dx = 2(5x^4) = 10x^4
The Product Rule
The product rule is used to find the derivative of products of two or more functions. To do this, you multiply the derivative of one function with the other function and add the reverse. Let u(x) and v(x) be two differentiable functions of x.
d(u*v)/dx = u*(dv/dx) + v*(du/dx)
Example: f(x) = (x^2)(x^3 – 4)
d(f(x))/dx = (x^2)(3x^2) + (x^3 – 4)(2x) = 3x^4 + 2x^4 – 8x = 5x^4 – 8x
The Quotient Rule
In calculus, the quotient rule is used when differentiating a function that is the quotient of two other functions, for which derivatives exist. This is done by multiplying the derivative of the numerator with the denominator and subtracting the reverse. Finally, divide by the denominator squared. Let g(x) and h(x) be two differentiable functions of x.
f(x) = g(x)/h(x)
d(f(x))/dx = [g*(d(h)/dx) – (d(g)/dx)*h]/(h)^2
Example: f(x) = (x-2)/x^2
d(f(x))/dx = [(x^2)(1) – (x-2)(2x)]/(x^2)^2 = (4x – x^2)/(x^4) = (4-x)/(x^3)
The Chain Rule
This rule is a formula used for determining the derivative of a composite function. In other words, if f is a function and g is a function, then the chain rule allows the derivation of the composite function f○g in terms of the derivatives of both f and g. Let f(x) and g(x) be two differentiable functions of x.
h(x) = f(g(x))
d(h(x))/dx = [d(f(g(x))/d(g(x))]*d(g(x))/dx
Example: j(x) = (7x+3)^2
d(j(x))/dx = 2*(x+3)*7 = 14x + 42
With these rules mastered, differentiation can be a breeze, but that doesn’t mean that first principles is useless. It’s an excellent way of understanding differentiation and can be used as a check when using the aforementioned rules.
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