Differentiation Formulas

Learn Differentiation Formulas

Differentiation is the process of finding the derivative of a function. The derivative is a measure of how a function changes as the input changes.

There are three basic differentiation formulas:

Derivative of a Constant

The derivative of a constant is simply the constant itself.

Derivative of a Power

The derivative of a power is the power multiplied by the derivative of the base.

Derivative of a Function

The derivative of a function is the function multiplied by the derivative of the input.

What is Differentiation?

Differentiation is the mathematical process of finding the derivative of a function. The derivative is a measure of how a function changes over time and is used to calculate rates of change.

Differentiation Rules:

The derivative of a function is a measure of the rate of change of the function with respect to a variable. The derivative is a function itself and is denoted by the symbol . The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

There are three basic rules of differentiation which can be used to differentiate any function.

• The derivative of a constant is zero.
• The derivative of a sum is the sum of the derivatives.
• The derivative of a product is the product of the derivatives.

Sum or Difference Rule:

The sum or difference rule states that if two functions are defined on the same set, and one is the sum or difference of the other, then they are both continuous on that set.

Product Rule:

The product rule states that the derivative of the product of two functions is the product of the derivatives of the individual functions.

Quotient Rule:

If f(x) and g(x) are integrable functions on an interval [a, b], and

f(x) = g(x) + c

Then:

f'(x) = g'(x) + c

Proof:

We can use the definition of the derivative to show that the statement is true.

We’ll start with the derivative of f(x):

f'(x) = lim h→0

[f(x + h) – f(x)] / h

We’ll also need the derivative of g(x):

g'(x) = lim h→0

[g(x + h) – g(x)] / h

Now, we can Substitute g(x) + c for f(x) and g(x) for f(x + h) in the first equation to get:

f'(x) = lim h→0

[g(x + h) – g(x)] / h

Now, we can multiply both sides of the equation by h to get:

f'(x) = lim h→0

[g(x + h) – g(x)]

Now, we can integrate both sides of the equation:

f'(x) = lim h→0

Chain Rule:

The chain rule states that the derivative of the composite function of two or more functions is the product of the derivatives of the individual functions.

For example, if is a composite function of and , then .

Differentiation Formulas:

• If y = x^3, then y’ = 3x^2
• If y = cos(x), then y’ = -sin(x)
• If y = x^2 + 2x, then y’ = 2x
• If y = 2x^3 – 5x, then y’ = 6x^2 – 10x

Differentiation Formulas for Trigonometric Functions:

• sin(x) = y
• cos(x) = x
• tan(x) = y/x
• cot(x) = 1/tan(x)
• sec(x) = 1/cos(x)
• csc(x) = 1/sin(x)

Differentiation Formulas for Inverse Trigonometric Functions:

• arcsin(x) = sin^{-1}(x)
• arccos(x) = cos^{-1}(x)
• arctan(x) = tan^{-1}(x)

Differentiation Formulas List:

1. y = mx + b
2. y = 2x
3. y = 3x
4. y = 5x
5. y = ax
6. y = bx
7. y = cx
8. y = dx
9. y = ex
10. y = ey

Power Rule:

If the derivative of a function is positive at a point, then the function increases at that point.

If the derivative of a function is negative at a point, then the function decreases at that point.

Sum Rule:

The sum rule states that the total number of protons and neutrons in an atom is constant.

Product Rule:

The product rule states that the derivative of the product of two functions is the product of the derivatives of the individual functions.

Quotient Rule:

• If ƒ(x) and g(x) are differentiable functions, then
• ƒ(x)g'(x) – ƒ(x)g(x)
• will be equal to the derivative of ƒ(x)g(x)

Other Differentiation Formulas:

1. If a quantity is multiplied by a negative number, the result is a negative quantity.

2. If a quantity is divided by a negative number, the result is a positive quantity.

3. If a quantity is raised to a negative power, the result is a negative quantity.

4. If a quantity is raised to a positive power, the result is a positive quantity.

Chain Rule:

If y = f(x) and y’ = f'(x), then

dy/dx = (dy/dt)(dt/dx)