Derivative Rules

The following diagram gives the basic derivative rules that you may find useful: Constant Rule, Constant Multiple Rule, Power Rule, Sum Rule, Difference Rule, Product Rule, Quotient Rule, and Chain Rule. Scroll down the page for more examples, solutions, and Derivative Rules.

 

How To Use The Differentiation Rules: Constant, Power, Constant Multiple, Sum And Difference?

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How To Use The Basic Derivative Rules? Constant Rule, Power Rule, Constant Multiple Rule, Exponential Rule, Sum And Difference Rule?

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Chain Rule For Finding Derivatives

a.k.a. the Outside-Inside Rule
How to use the chain rule to find the derivative and when to use it.

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1. Constant Rule:
   The derivative of a constant is always zero.
   If f(x) = c (where c is a constant), then f’(x) = 0.
   Example: If f(x) = 5, then f’(x) = 0.
   

2. Power Rule:
   This rule is for functions of the form f(x) = xⁿ.
   If f(x) = xⁿ, then f’(x) = n x^(n-1).
   Examples:
   If f(x) = x⁷, then f’(x) = 7x⁶.
   If f(x) = x^-2, then f’(x) = -2x^-3.
   

3. Constant Multiple Rule:
   If you have a constant multiplied by a function, you can take the constant outside the derivative.
   If f(x) = c g(x) (where c is a constant), then f’(x) = c g’(x).
   Example: If f(x) = 4x², then f’(x) = 4 (2x) = 8x.
   

4. Sum/Difference Rule:
   The derivative of a sum or difference of functions is the sum or difference of their derivatives.
   If h(x) = f(x) + g(x), then h’(x) = f’(x) + g’(x).
   If h(x) = f(x) - g(x), then h’(x) = f’(x) - g’(x).
   Example: If h(x) = x³ + 2x, then h’(x) = 3x² + 2.
   

5. Product Rule:
   The derivative of the product of two functions is given by:
   If h(x) = f(x) g(x), then h’(x) = f’(x)g(x) + f(x)g’(x).
   

6. Quotient Rule:
   The derivative of the quotient of two functions is given by:
   If h(x) = f(x) / g(x), then h’(x) = [f’(x)g(x) - f(x)g’(x)] / [g(x)]².
   

7. Chain Rule:
   This rule is for composite functions (functions within functions).
   If h(x) = f(g(x)), then h’(x) = f’(g(x)) g’(x).
   “Derivative of the outside times derivative of the inside."
   Example: If h(x) = (x² + 1)³, then h’(x) = (3(x² + 1)²)(2x) = 6x(x² + 1)².

These are the fundamental derivative rules. There are other rules for specific functions (like trigonometric, exponential, and logarithmic functions), but these are built upon these basic rules. Learning and practicing these rules is essential for calculus.

The following diagram gives some derivative rules that you may find useful for Exponential Functions, Logarithmic Functions, Trigonometric Functions, Inverse Trigonometric Functions, Hyperbolic Functions, and Inverse Hyperbolic Functions.

Besides the basic rules (power rule, product rule, etc.), there are specific derivative rules for certain types of functions.
Here are the most common ones:

1. Trigonometric Functions:
   Derivative of sin(x): d/dx [sin(x)] = cos(x)
   Derivative of cos(x): d/dx [cos(x)] = -sin(x)
   Derivative of tan(x): d/dx [tan(x)] = sec²(x)
   Derivative of cot(x): d/dx [cot(x)] = -csc²(x)
   Derivative of sec(x): d/dx [sec(x)] = sec(x)tan(x)
   Derivative of csc(x): d/dx [csc(x)] = -csc(x)cot(x)
   

2. Exponential Functions:
   Derivative of e^x: d/dx [e^x] = e^x
   Derivative of a^x (where a is a constant > 0 and a ≠ 1): d/dx [a^x] = a^x ln(a)
   

3. Logarithmic Functions:
   Derivative of ln(x): d/dx [ln(x)] = 1/x
   Derivative of log_a(x): d/dx [log_a(x)] = 1/(x ln(a))
   

4. Inverse Trigonometric Functions:
   Derivative of arcsin(x) (or sin^-1(x)): d/dx [arcsin(x)] = 1 / √(1 - x²)
   Derivative of arccos(x) (or cos^-1(x)): d/dx [arccos(x)] = -1 / √(1 - x²)
   Derivative of arctan(x) (or tan^-1(x)): d/dx [arctan(x)] = 1 / (1 + x²)
   

Derivatives Of Exponential Functions

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Derivatives Of Logarithmic Functions

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Derivative Problems Involving Trigonometric Functions

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Inverse Trigonometric Functions - Derivatives

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Hyperbolic Functions - Derivatives

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Inverse Hyperbolic Functions - Derivatives

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