Differentiation, Integration and Centroids

Differentiation (common derivatives):

d/dx( c )= 0

The derivative of a constant is zero.

Example: d/dx 7 = 0

d/dx( c × x )= c

The rate of change of a linear function is its slope.

Example: d/dx 3 × x = 3

d/dx (xⁿ) = n × x^(n-1)

Example: d(x⁴)/dx  = 4 × x ³

d/dx (log x) = 1/x

The derivative of the log of x is its inverse.

Example: d(log (x + 1))/dx  = 1 / (x + 1)

d/dx (e^ax) = a e^ax

Example: d (e^3x) /dx= 3 e^3x

d/dx (sin cx) = c cos x

Example: d(sin3x) /dx = 3cos x

d/dx (cos x) = -sin x

Example: d (cos ) /dx= - sin

Integral of a function: The integral of a function f(x) over an interval from x₁ to x₂ yield the area under the curve in this interval

Note: The integral represents the  as

Some indefinite integrals to remember:

Note: Remember to add a constant of integration if you are not specifying limits. You evaluate the constant of integration by forcing the integral to pass through a known point.

Note: For definite integrals subtract the value of the integral at the lower limit from its value at the upper limit. For example, if you have the indefinite integral.

Note: The following notation is common

Integration by parts:

Centroid of an area:

The centroid of an area is the area weighted average location of the given area.

Centroids of common shapes:

Some other centroids of common shapes of areas and lines are as follow:

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