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Calculus | Derivatives of a Function – Lesson 7 | Don’t Memorise
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Derivatives of a function measures its instantaneous rate of change. It also tells us the slope of a tangent line at a point on the curve (graph of the function).
Watch this video to understand what the derivative of a function is and how to find it.
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In this video, we will learn:
0:00 Which is the Hardest Mountain to Climb in the World?
0:36 Steepness
1:50 Tangent Function
2:29 Derivatives of a Function
8:31 Instantaneous Rate of Change
9:01 Average Speed
10:16 Instantaneous Speed
10:36 instantaneous Rate of Change of a Function
Watch the Calculus Derivative of a Function Lesson 8 here https://www.youtube.com/watch?v=AOkn9UK5AU
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Derivatives DerivativeOfAFunction Differentiation
Basic Differentiation Rules For Derivatives
This calculus video tutorial provides a few basic differentiation rules for derivatives. It discusses the power rule and product rule for derivatives. It also explains how to find the derivative of trigonometric functions, exponential functions, and natural logarithmic functions.
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Derivatives of Logarithmic Functions:
https://www.youtube.com/watch?v=Dp9sgIvaKPk
Logarithmic Differentiation:
https://www.youtube.com/watch?v=lFl8ekyr63U
Implicit Differentiation:
https://www.youtube.com/watch?v=XQDh6Z6DPI
Quotient Rule For Derivatives:
https://www.youtube.com/watch?v=8jVDEcQ0wXk
Chain Rule For Derivatives:
https://www.youtube.com/watch?v=HaHsqDjWMLU
Limits and Derivatives:
https://www.youtube.com/watch?v=aTLjoDT1GQ\u0026t=1s
How To Find The Equation of The Tangent Line:
https://www.youtube.com/watch?v=UOrS2qje2_o
[TOÁN CAO CẤP – CHUYÊN ĐỀ 12] BÀI 12.1 – ĐẠO HÀM RIÊNG CẤP 1 (THE FIRST ORDER PARTIAL DERIVATIVES)
ĐẠO HÀM RIÊNG CẤP 1 CỦA HÀM 2 BIẾN, THE FIRST ORDER PARTIAL DERIVATIVES
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The Chain Rule… How? When? (NancyPi)
MIT grad shows how to use the chain rule to find the derivative and WHEN to use it. To skip ahead: 1) For how to use the CHAIN RULE or \”OUTSIDEINSIDE rule\”, skip to time 0:17. 1b) For how to know WHEN YOU NEED the chain rule, skip to 4:35. 2) For another example with the POWER RULE in the chain rule, skip to 7:05. 3) For a TRIG derivative chain rule example, skip to 9:33. 3b) For the formal chain rule FORMULA, skip to 11:36. PS) For a DOUBLE CHAIN RULE (or \”repeated use of the chain rule\”) example, skip to 13:33. Nancy formerly of MathBFF explains the steps.
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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 \”OUTSIDEINSIDE\” 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) TRIG EXAMPLE: the idea is the same as above even if you’re using the chain rule to differentiate something like a trigonometric function. If you have anything more than just x inside the trig function, you’ll need the chain rule to find the derivative. For the equation y = sin(x^2 3x), you first take the derivative of the outer function, just the sine function. Since the derivative of sine is cosine, the outside derivative (with the inside left unchanged) is cos(x^2 3x). Then, find the derivative of just the inside (of just the x^2 3x part), and multiply by that. Since the derivative of x^2 3x is 2x 3, the full derivative answer is dy/dx = cos(x^2 3x)(2x 3).
3b) FORMULA: Although it’s easier to think about the chain rule as the \”outsideinside 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.
P.S.) DOUBLE CHAIN RULE: Sometimes you might have to use the chain rule more than once, known as \”repeated use of the chain rule\”. In y = (1 + cos2x)^2, not only would you need to take the derivative of the outside power of 2, as well as multiply by the derivative of the inside function, 1 + cos2x, but after that you would ALSO then need to multiply by the derivative of the 2x inside cosine because that inside function was 1 + cos2x and not just 1 + cosx. This means you would use the chain rule twice. The idea is that you have to keep taking the derivative of the inner functions until you have reached every inner function that is more complicated than just \”x\”.
For more calculus math videos, check out: http://nancypi.com
Derivatives of Exponential Functions
This calculus video tutorial explains how to find the derivative of exponential functions using a simple formula. It explains how to do so with the natural base e or with any other number. This video contains plenty of examples and practice problems including those using the product rule and quotient rule for derivatives.
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