A function expresses relationships between constants and one or more variables. For example, the function f(x) = 5x + 10 expresses a relationship between the variable x and the constants 5 and 10. Known as derivatives and expressed as dy/dx, df(x)/dx or f’(x), differentiation finds the rate of change of one variable with respect to another -- in the example, f(x) with respect to x. Differentiation is useful for finding the optimal solution, meaning finding the maximum or minimum conditions. Some basic rules exist with regard to differentiating functions.

Differentiate a constant function. The derivative of a constant is zero. For example, if f(x) = 5, then f’(x) = 0.

Apply the power rule to differentiate a function. The power rule states that if f(x) = x^n or x raised to the power n, then f'(x) = nx^(n - 1) or x raised to the power (n - 1) and multiplied by n. For example, if f(x) = 5x, then f'(x) = 5x^(1 - 1) = 5. Similarly, if f(x) = x^10, then f'(x) = 9x^9; and if f(x) = 2x^5 + x^3 + 10, then f'(x) = 10x^4 + 3x^2.

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Find the derivative of a function using the product rule. The differential of a product is not the product of the differentials of its individual components: If f(x) = uv, where u and v are two separate functions, then f'(x) is not equal to f'(u) multiplied by f'(v). Rather, the derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first. For example, if f(x) = (x^2 + 5x) (x^3), the derivatives of the two functions are 2x + 5 and 3x^2, respectively. Then, using the product rule, f'(x) = (x^2 + 5x) (3x^2) + (x^3) (2x + 5) = 3x^4 + 15x^3 + 2x^4 + 5x^3 = 5x^4 + 20x^3.

Get the derivative of a function using the quotient rule. A quotient is one function divided by another. The derivative of a quotient equals the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, then divided by the denominator squared. For example, if f(x) = (x^2 + 4x) / (x^3), the derivatives of the numerator and the denominator functions are 2x + 4 and 3x^2, respectively. Then, using the quotient rule, f'(x) = [(x^3) (2x + 4) - (x^2 + 4x) (3x^2)] / (x^3)^2 = (2x^4 + 4x^3 - 3x^4 - 12x^3) / x^6 = (-x^4 - 8x^3) / x^6.

Use common derivatives. The derivatives of common trigonometric functions, which are functions of angles, need not be derived from first principles -- the derivatives of sin x and cos x are cos x and -sin x, respectively. The derivative of the exponential function is the function itself -- f(x) = f’(x) = e^x, and the derivative of the natural logarithmic function, ln x, is 1/x. For example, if f(x) = sin x + x^2 - 4x + 5, then f'(x) = cos x + 2x - 4.