After all, finding the derivative of 25 is something you can probably do in an instant. You may be wondering why I split this up into three parts. Step 3: Place your answers from the left hand side (Step 1) and right hand side (Step 2) back together: Step 2: Take the derivative of the right hand side. We have two parts to differentiate: x 2 and 4 y 2:Ĭombining the answers (because of the Sum Rule) we get: Step 1: Take the derivative(s) of the left hand side. Part One: Finding the First Derivative Implicitly Finding the Second Derivative Implicitly: ExampleĮxample question: Find the second derivative implicitly of x 2 + 4y 2 = 1. For example, to take the derivative of an expression like 4 y 2 with respect to x, you have an inside function and an outside function: The key to finding the second derivative implicitly requires a good understanding of the chain rule. Like the “usual” way of finding second derivatives, finding the second derivative implicitly involves two steps: implicitly differentiating twice. Note: You can’t always take the second derivative of a function. ExamplesĮxample question 1: Find the 2nd derivative of 2x 3. This is useful when it comes to classifying relative extreme values if you can take the derivative of a function twice you can determine if a graph of your original function is concave up, concave down, or a point of inflection. One reason to find a 2nd derivative is to find acceleration from a position function the first derivative of position is velocity and the second is acceleration. In other words, in order to find it, take the derivative twice. The second derivative (f ′′), is the derivative of the derivative (f ′).
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