Derivatives are fundamental in calculus, measuring how fast a function changes at any given point. Think of velocity as the derivative of position - it tells us the instantaneous rate of change. Geometrically, the derivative at a point equals the slope of the tangent line to the curve at that point. Let's see this with the function f of x equals x squared, where the derivative is 2x.
Now let's explore the basic derivative rules that form the foundation of differentiation. The power rule states that the derivative of x to the n is n times x to the n minus 1. For example, the derivative of x cubed is 3 x squared. The constant rule tells us that the derivative of any constant is zero. The sum rule allows us to differentiate each term separately. These rules can be combined to handle complex polynomial functions efficiently.
When dealing with products and quotients of functions, we need specialized rules. The product rule states that the derivative of f times g equals f prime times g plus f times g prime. For example, the derivative of x squared sine x is 2x sine x plus x squared cosine x. The quotient rule handles fractions: the derivative of f over g equals f prime g minus f g prime, all over g squared. Let's apply this to x squared over x plus 1, giving us x squared plus 2x over x plus 1 squared.
The chain rule is essential for differentiating composite functions, where one function is nested inside another. The rule states that the derivative of f of g of x equals f prime of g of x times g prime of x. We use the outside-inside method: first differentiate the outside function, then multiply by the derivative of the inside function. For sine of x squared, we get cosine of x squared times 2x. For 3x plus 1 to the fifth power, we get 5 times 3x plus 1 to the fourth times 3, which simplifies to 15 times 3x plus 1 to the fourth.
Implicit differentiation is used when y is not explicitly solved for x, such as in equations like x squared plus y squared equals 25. We differentiate both sides with respect to x, treating y as a function of x and applying the chain rule to y terms. For the circle equation, we get 2x plus 2y dy dx equals zero, solving to get dy dx equals negative x over y. For more complex equations like xy plus y squared equals 10, we collect all dy dx terms and factor them out to solve for the derivative.