Common Derivatives

Derivative of a function of a real variable measures the sensitivity to change of the function value (output) with respect to a change in its argument (input).

Derivatives are a fundamental tool of calculus. The derivative of a function of a single variable at a chosen input value (when it exists) is the slope of the tangent line to the graph of the function at that point.

The process of finding a derivative is called differentiation. The reverse process of differentiation is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration.

Differentiation and integration constitute the 2 fundamental operations in single-variable calculus.

Common Function

The derivative of a function y = f(x) of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x.

It is called “the derivative of f with respect to x”. If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point.

Review the differentiation rules for all the common function types.

 FunctionDerivative
PolynomialsCommon Derivaties PolynomialCommon Derivaties Polynomial Function
RadicalsSin InverseSin Inverse Derivative
Trigonometric functionssin(x)cos(x)
 cos(x)– Sin(x)
 tan(x)Sec Square
 cot(x)Cot Inverse
 sec(x)Sec Inverse
 csc(x)Cosec Inverse
Inverse trigonometric functionsSin InverseSin Inverse Derivative
 Cos InverseCos Inverse Derivative
 Tan InverseTan Inverse Derivative
Exponential functionsExponential FunctionExponential Function
 Exponential Function aExponential Function a Inverse
Logarithmic functionsLogarithmic FunctionLogarithmic Function Inverse
 Logarithmic Function bLogarithmic Function b Inverse

Examples

 FunctionApplicationDerivative
Polynomials2x^323x^36^2
RadicalsExample RadicalExample Radical 2Example Radical 3
Trigonometric functionssin(x)cos(x)cos(x)
 cos(x)– sin(x)– sin(x)
 tan(x)Sec SquareExample Trigonometric
 cot(x)Cot InverseExample Trigonometric 2
 sec(x)Sec InverseExample Trigonometric 3
 csc(x)Cosec InverseExample Trigonometric 4
Inverse trigonometric functionsSin InverseSin Inverse DerivativeSin Inverse Derivative
 Cos InverseCos Inverse DerivativeCos Inverse Derivative
 Tan InverseTan Inverse DerivativeTan Inverse Derivative
Exponential functionsExponential FunctionExponential FunctionExponential Function
 Example ExponentialExample Exponential 2Example Exponential 3
Logarithmic functionsLogarithmic FunctionLogarithmic Function InverseLogarithmic Function Inverse
 Example LogarithmicExample Logarithmic 2Example Logarithmic 3

Derivative rules

The derivative of a function can be computed from the definition by considering the difference quotient & computing its limit.

Once the derivatives of some simple functions are known, the derivatives of other functions are computed more easily using rules for obtaining derivatives of more complicated functions from simpler ones.

Here are simple rules to help you work out the derivatives of some functions.

RulesFunctionDerivative
Multiplication by constantcfcf’
Power Rulexnnxn-1
Sum Rulef + gf’ + g’
Difference Rule f – gf’ – g’
Product Rule f gf g’ + f’ g
Quotient Rulef/g(f g’ – f’ g)/g2
Chain Rule f (g(x)) f’(g(x)) g’(x)
Reciprocal Rule1/f -f’/f2

Examples

RulesFunctionApplicationDerivative
Multiplication by constant7(2x3)7 × (2 × 3 × x3-1)42x2
Power Rulex77x7-17x6
Sum Rulex7 + x67x7-1 + 6x6-17x6 + 6x5
Difference Rule X5  x65x5-1  6x6-15x4 + 6x5
Product Rule sin(x) cos(x)sin(x)(-sin(x)) – cos(x)cos(x)– sin2(x) + cos2(x)
Chain Rulesin(x3) cos(x3)×3x23x2cos(x3)
Reciprocal Rule Logarithmic Function Inverse Example Reciprocal 1Example Reciprocal 2

Learn More

Differentiable

Implicit Differentiation

Integration

Integration by Substitution (Reverse Chain Rule)

Calculus Index

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