Differentiation via Limit Process and Chain Rule Applications
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The derivative of a function is defined as the limit of the difference quotient as h approaches zero. This gives us the instantaneous rate of change at any point. Let's see how this works with our example function f of x equals 3 x squared times e to the power of 2 x cubed minus 1. We can visualize this as the slope of the secant line approaching the slope of the tangent line.
Now let's work through the differentiation step by step. First, we apply the limit definition. Then we substitute f of x plus h into our formula. Since our function is a product of 3 x squared and e to the power of 2 x cubed minus 1, we'll need to use the product rule. The derivative will be the sum of the derivative of the first factor times the second factor, plus the first factor times the derivative of the second factor.
When dealing with composite functions, we apply the chain rule. The chain rule states that the derivative of f of g of x equals f prime of g of x times g prime of x. Let's see this in action with two examples. First, for cosine of the natural log of 5x to the fourth power, we differentiate the outer function cosine, then multiply by the derivative of the inner function. For the logarithm base 5 of x over x minus 1, we again apply the chain rule step by step.
For products and quotients of functions, we use the product rule and quotient rule. The product rule states that the derivative of u times v equals u prime times v plus u times v prime. The quotient rule for u over v equals u prime v minus u v prime, all over v squared. Let's apply these to our examples. For 2 to the x times sine x, we use the product rule. For cotangent x times x plus 4 to the two thirds power, we again apply the product rule with careful attention to the derivatives of each factor.