Differentiation is a method to compute the rate at which a dependent output y changes
with respect to the change in the independent input x. This rate of change is called
the
derivative of y with respect to x.
In other words,
The process of finding a derivative is called Differentiation.
The graph of a function, drawn in blue, and a tangent line to that function, drawn in green.
The slope of the tangent line is equal to the derivative of the function at the marked point.
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Derivative of a constant function
The derivative of f(x) = c where c is a constant is given by
f '(x) = 0
Example: f(x) = - 10 , then f '(x) = 0
Derivative of a power function (Power rule)
The derivative of f(x) = x
r where r is a constant real number is given by
f '(x) = r *x(r-1)
Example: f(x) = x
-2 , then f '(x) = -2 x
-3 = -2 / x
3
Derivative of a function multiplied by a constant
The derivative of f(x) = c g(x) is given by
f '(x) = c g '(x)
Example: f(x) = 3x
3 ,
let c = 3 and g(x) = x
3, then f '(x) = c g '(x)
= 3 (3x
2) = 9 x
2
Derivative of the sum of functions (Sum rule)
The derivative of f(x) = g(x) + h(x) is given by
f '(x) = g '(x) + h '(x)
Example:
f(x) = x
2 + 4
let g(x) = x
2 and h(x) = 4, then f '(x) = g '(x) + h '(x) = 2x + 0 = 2x
Derivative of the difference of functions
The derivative of f(x) = g(x) - h(x) is given by
f '(x) = g '(x) - h '(x)
Example:
f(x) = x
3 - x
-2
let g(x) = x
3 and h(x) = x
-2, then
f '(x) = g '(x) - h '(x) = 3 x
2 - (-2 x
-3) = 3 x
2 + 2x
-3
Derivative of the product of two functions(Product rule)
The derivative of f(x) = g(x) h(x) is given by
f '(x) = g(x) h '(x) + h(x) g '(x)
Example:
f(x) = (x
2 - 2x) (x - 2)
let g(x) = (x
2 - 2x) and h(x) = (x - 2), then
f '(x) = g(x) h '(x) + h(x) g '(x) = (x
2 - 2x) (1) + (x - 2) (2x - 2)
= x
2 - 2x + 2 x
2 - 6x + 4
=3 x
2 - 8x + 4
Derivative of the quotient of two functions (Quotient rule)
The derivative of f(x) = g(x) / h(x) is given by
f '(x) = [ h(x) g '(x) - g(x) h '(x) ] / h(x)2
Example: f(x) = (x - 2) / (x + 1)
let g(x) = (x - 2) and h(x) = (x + 1), then
f '(x) = ( h(x) g '(x) - g(x) h '(x) ) / h(x)
2
= ( (x + 1)(1) - (x - 2)(1) ) / (x + 1)
2
= 3 / (x + 1)
2
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An equation of the form f(x, y) in which y is not expressible directly in terms of x, is known as an implicit function of x and y.
To implicitly differentiate an implicit function follow the steps below.
1. Apply all differentiation rules. Except to y.
Given equation is:
y
3 - x
2 = -5
y3 - 2x = 0
2. Then differentiate the y's in regard to x. In turn you add a y prime
to the differentiated y's
3y
2y' -2x = 0
Here, y' is refered to d(y)/d(x)
3. Now factor out any y primes
Hint: In this example there are none.
4. Lastly, place all the y primes on one side
Solution:
y' = (2x) / (3y2)
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