Differentiation

• Differentiation is a process in calculus that finds the rate at which a quantity is changing at a given point.
• The derivative, often represented as dy/dx or f’(x), is the result of differentiation.
• If a function y = f(x) is differentiated, its derivative represents the slope (or gradient) of the tangent to the curve of the function at any given point.
• You can determine a function’s derivative using specific rules such as the power rule, product rule, quotient rule, or chain rule.
• The power rule: if y = ax^n, then dy/dx = nax^(n−1).
• The product rule: it’s used when differentiating functions that are being multiplied together. The derivative is the first function times the derivative of the second function plus the second function times the derivative of the first function.
• The quotient rule: if you’re differentiating a function that divides two functions, this rule applies. It’s the bottom function times the derivative of the top function minus the top function times the derivative of the bottom function, all divided by the bottom function squared.
• The chain rule: it’s used when differentiating composite functions. In this, you find the derivative of the outside function, then multiply it by the derivative of the inside function.
• If the derivative is positive for a certain interval on the function, it means the function is increasing (going upwards) in that interval. If the derivative is negative, the function is decreasing (going downwards).
• When the derivative equals zero, it may indicate a local maximum or minimum, these are often called ‘turning points’.
• Differentiation can be applied in various real-world situations such as optimizing a function to find the highest or lowest value, and understanding how change in one variable affects another.