Glossary
Chain Rule
A rule used to find the derivative of a composite function, where one function is nested inside another.
Example:
To differentiate h(x) = cos(x^2), you must use the Chain Rule because x^2 is inside the cosine function.
Coefficient
A numerical or constant quantity placed before and multiplying the variable in an algebraic expression.
Example:
In the expression 3x^5, the coefficient is 3, which gets multiplied by the exponent when differentiating.
Constant (in derivatives)
A fixed numerical value that does not change; its derivative is always zero.
Example:
When differentiating h(x) = 4x^3 + 9, the constant 9 differentiates to 0.
Derivative
The instantaneous rate of change of a function, representing the slope of the tangent line to the function's graph at any given point.
Example:
The derivative of a car's position function tells you its instantaneous velocity.
Exponent
A number indicating the power to which a base number or variable is raised.
Example:
In the term x^4, the exponent is 4, which is crucial for applying the Power Rule.
Fractions (in differentiation)
Rational expressions where variables are in the denominator, which must be rewritten using negative exponents before applying the Power Rule.
Example:
To differentiate , you first rewrite the fraction as x^(-2).
Power Function
A function of the form f(x) = kx^n, where 'k' and 'n' are constants.
Example:
The function g(x) = 5x^(-2) is a power function that can be easily differentiated using the Power Rule.
Power Rule
A fundamental rule for finding the derivative of functions in the form of x^n, where the exponent 'n' is brought down as a coefficient and then reduced by one.
Example:
To find the derivative of f(x) = x^7, you apply the Power Rule to get f'(x) = 7x^6.
Product Rule
A rule used to find the derivative of a function that is the product of two or more differentiable functions.
Example:
To differentiate f(x) = (x^2 + 1) * sin(x), you would need to apply the Product Rule.
Quotient Rule
A rule used to find the derivative of a function that is the ratio of two differentiable functions.
Example:
When finding the derivative of g(x) = , the Quotient Rule is essential.
Radicals (in differentiation)
Expressions involving roots (like square roots or cube roots) that must be rewritten as fractional exponents before applying the Power Rule.
Example:
Before differentiating , you must rewrite the radical as x^(1/2).
Tangent Line
A straight line that touches a curve at a single point and has the same slope as the curve at that point.
Example:
The derivative of a function at x=2 gives the slope of the tangent line to the function's graph at that specific point.