![]() ![]() Instead, it would be more efficient to split it into parts by combining aĬouple of rules of differentiation together. Separate components, but this would require a large number of calculations. Since it is a polynomial function, it would be possible for us to expandĪll the parentheses via multiplication and take the derivatives of the Let us first analyze the given function and see what rules we can apply to We can represent this visually as follows.Įxample 1: Finding the First Derivative of Polynomial Functions at a Point Using theįind the first derivative of □ = ( □ − 5 ) ( □ − 2 ) at ( 1, − 4 ). Continuing this process, we can continue to remove layers of complexityįrom the function until we arrive at elementary expressions that we know how toĭifferentiate. □ ( □ ), but this too can be broken down into smaller ![]() This ofĬourse means that we need to find the derivative of In other words, □ ( □ ) is a composition ofįunctions, so we can apply the chain rule to help us differentiate it. □ ( □ ) separately in this way, we can see that Separately and add them together afterwards. Using the linearity of differentiation, this means we can differentiate If we do this, we can see that it is the sum Generally, the best way to do this is to begin by considering the outermost Īt first, this may seem impossible to deal with, but we can break it into parts. Many different operations together, and how we can tackle the differentiation by Let us consider an example of differentiating a complicated function combining Will look at a number of examples that will highlight the skills we need Simplifications that will make the process easier. Trivial exercise and it can be challenging to identify the correct rules toĪpply, the best order to apply them, and whether there are algebraic However, we should be aware that this is often not a In addition to using these rules separately, it is also possible to use them inĬonjunction with each other, allowing us to differentiate any combination ofĮlementary functions. For differentiable functions □ ( □ ) andĪnd constants □, □ ∈ ℝ, we have the following rules: ![]()
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