This is very easy to do by replacing different occurrences of $x$ with separate variables, computing the partial derivatives, adding them up and setting all the variables to the same value $x$. To understand why the above technique is useful try to compute the derivative of functions such as $f(x)=x^x$. To see why this is the case, we consider an example involving meaningful functions. For me a part of being intuitive is the ability of immediately detect pattern and use it in other circumstances, and this approach goes well beyond the product rule. The product rule As part (b) of Preview Activity 2.3.1 shows, it is not true in general that the derivative of a product of two functions is the product of the derivatives of those functions. $d(g\cdot h)(x,x) = h\cdot g'(x) \cdot dx + g\cdot h'(x) \cdot dx$ĭifferent people have different notions about what is intuitive. $d(g\cdot h)(y,z) = h(z)\cdot g'(y) \cdot dy + g(y)\cdot h'(z) \cdot dz$įinally, it remains to consider what happens when both $y$ and $z$ have the same value $x$: Because $g(y)$ is constant with respect to $z$ and $h(z)$ is constant with respect to $y$ and differentiation is linear we have: Now suppose that $f$ splits into a product of two functions, each being a function of just one of the variables: $f=g(y)\cdot h(z)$. The above is just a generalization of the chain rule, and IMO is very intuitive. The product and quotient of functions rules follow exactly the same logic: hold all variables constant except for the one that is changing in order to determine the slope of the function with respect to that variable. $df=\partial f/\partial y \cdot dy + \partial f/\partial z \cdot dz$ Combine the differentiation rules to find the derivative of a polynomial or rational function. Extend the power rule to functions with negative exponents. Worked example: Product rule with mixed implicit & explicit. Use the quotient rule for finding the derivative of a quotient of functions. Course: AP®/College Calculus AB > Unit 2. Specifically for the product rule, take a function of two variables $f(y,z)$ and consider the formula for the differential of $f$: Use the product rule for finding the derivative of a product of functions. The product rule tells us the derivative of two functions f and g that are multiplied together: (fg)’ fg’ + gf’ (The little mark ’ means 'derivative of'. In this example they both increase making the area bigger.For me intuition for product rule, as well as a couple of other techniques, comes from multi-variable calculus. The derivative is the rate of change, and when x changes a little then both f and g will also change a little (by Δf and Δg). When we multiply two functions f(x) and g(x) the result is the area fg: It is a rule that states that the derivative of a product of two functions is equal to the first function f(x) in its original form multiplied by the derivative of the second function g(x) and then added to the original form of the second function g(x) multiplied by the derivative of the first function f(x). (cos(x)sin(x))’ = cos(x) cos(x) + −sin(x) sin(x)Īnswer: the derivative of cos(x)sin(x) = cos 2(x) − sin 2(x) Why Does It Work? Use the quotient rule to show that is decreasing. (cos(x)sin(x))’ = cos(x) sin(x)’ + cos(x)’ sin(x) Use the product rule to show that f ( x ) 2 is also increasing. We have two functions cos(x) and sin(x) multiplied together, so let's use the Product Rule: Example: What is the derivative of cos(x)sin(x) ?
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