Calculus - Differentiating Using the Chain Rule

This calculus video tutorial give a quick refresh on how to find derivatives using the chain rule in an exam style situation. 1) The CHAIN RULE is one of the derivative rules. You need it to take the derivative when you have a function inside a function, or a "composite function". For ex, in the equation y = (3x + 1)^7, since the function 3x+1 is inside a larger, outer function, the power of 7, you'll need the chain rule to find the correct derivative. How do you use the chain rule? You can think of it as the "OUTSIDE-INSIDE" rule: take the DERIVATIVE of JUST the OUTSIDE function first, LEAVING THE INSIDE FUNCTION alone (unchanged), then MULTIPLY BY the DERIVATIVE of JUST the INSIDE function. Sometimes you might hear this expressed as: take the derivative of the outer function, "evaluated at the inner function", times the derivative of just the inner function. For our ex, first take the derivative of the outer function (the power of 7) to get 7*(3x + 1)^6 since the derivative "power rule" tells you to bring down the power to the front (as a constant or coefficient just multiplied in the front) and then decrease the power by 1, which leaves a power of 6. Notice that you leave the inside function the way it is and just rewrite it for now. Then you multiply by the derivative of just the inner function, 3x + 1. Since the derivative of 3x + 1 is just 3, the full derivative (dy/dx) is: 7*[(3x + 1)^6]*3, which is just 21(3x + 1)^6. 1b) HOW do you know WHEN TO USE the chain rule? If the original equation had just been x^7, there would be no need for the chain rule. It's when you have something more than just x inside that you should use the chain rule, such as (3x + 1)^7 or even (x^2 + 1)^7. Sometimes the chain rule may make no difference. For instance, if you have the function (x + 1)^7, taking the derivative of the inside function just gives you 1, so multiplying by that inside derivative of 1 will not change the overall answer. However, it can't hurt to use the chain rule anyway, so it's a good idea to get in the habit of using it so that you don't forget it when it really does make a difference. 2) Another chain POWER RULE example: To find the derivative of h(x) = (x^2 + 5x - 6)^9, use the same steps as above to first take the outside derivative and then multiply by the inside derivative. In this case, the derivative, dh/dx (or h'(x)) is equal to 9(x^2 + 5x - 6)^8 * (2x + 5). Using the chain rule with the power rule is sometimes called the "power chain rule". 3) FORMULA: Although it's easier to think about the chain rule as the "outside-inside rule", if for any reason you have to use the formal chain rule formula, check out the two versions I show here. Both are based on the equation being a composition of functions, f(g(x)). The second version shown uses Liebniz notation. Either way, both show a component of the derivative that comes from the inside function, and it's important not to forget to multiply by this inside derivative factor if you want to get the right full derivative answer. My social contacts: My Instagram Account: https://www.instagram.com/harrywalden/ My Youtube Channel: https://www.youtube.com/HaroldWalden My Facebook Page: https://www.facebook.com/claremont.tuition/ My LinkedIn Page: https://www.linkedin.com/haroldwalden/