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Integration and Differentiation
Transcript of Integration and Differentiation
and why do
we need it? Differentiation Differentiation is primarily used for calculating rates of change, and more specifically to find the gradient at a specific point on a graph. In Mechanics, the rate of change of displacement (with respect to time) is the velocity and the rate of change of velocity (with respect to time) is the acceleration. When illustrating a function on a graph the rate of change is represented by the gradient. Chain Rule Let's use this equations as an example:
y = (3x − 2)4
If we let u = 3x − 2 to give us y = u4.
We can now differentiate this to give us:
dy/du = 4u^3
Integration and Differentiation Methods include:
Integration by parts
Substitution Methods include:
Chain rule If we start with the equation:
ʃ ln x dx
we need to label each part to complete the calculation:
uv - ʃ vdu Integration by Parts From this we can multiply out the brackets:
x ln x - ʃ dx
And we now have our answer:
x ln x - x Integration by Parts We can let:
u = lnx
du = 1 / x
dv = dx
v = x
Therefore substituted into the equation:
ln(x) - ʃ x (1/x) dx Chain Rule The only problem is that we want dy/dx, not dy/du, and this is where we use the chain rule. The chain rule states that:
dy/dx = (dy/du) * (du/dx)
As we already know u and du/dx, we can substitute them into the equation as follows:
du/dx = (4u^3) * 3 = 12 (3x - 2)^3
And that's our answer!