The chain rule allows us to compute the derivative of a composite function. In other words suppose that:
for some functions f and g. For example:
In this case we can identify:
How do you compute the derivative in this case? The chain rule tells us that we compute the derivative in the following way:
Let’s compute some examples. [...]
Posted on March 8th, 2010 by AdamP
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In this video integrals involving powers of trig functions are illustrated. Usually this kind of integration involves some form of either u substitution or forces us to call upon various trig identities. See our earlier post on trig integrals for more discussion and examples.
Posted on March 8th, 2010 by AdamP
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Implicit differentiation is a technique used to compute a derivative when a function y = f(x) is given indirectly by an equation that relates x and y. For example suppose that you are given:
The equation must be solved for y to obtain an explicit solution, and in this case there are two solutions. The procedure [...]
Posted on March 8th, 2010 by AdamP
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Doing trig integrals is usually pretty simple. This is because of the cyclic nature of the derivatives of trig functions. Let’s recall that:
From here we can write down their anti-derivatives:
The minus sign comes about because we integrated the derivative of cos x to get this result. That is
but this was equal to minus sin x. [...]
Posted on March 8th, 2010 by AdamP
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Solving an integral with u substitution brings an integral into elementary form through a change of variable technique. In this example we consider a problem where it may not be obvious how to apply the method. Consider:
We’ll see in a moment that this integral evaluates to an inverse tangent. We’re going to use two steps [...]
Posted on March 7th, 2010 by AdamP
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