Example Problem 3. Chain Rule for Derivatives

p {margin: 0; padding: 0;} .ft00{font-size:19px;font-family:CourierNew;color:#000000;} .ft01{font-size:16px;font-family:CourierNew;color:#000000;} .ft02{font-size:16px;line-height:18px;font-family:CourierNew;color:#000000;} Chain Rule for Derivatives. Example Problem 3 Let's take the derivative of the cosine, let's do cosine to the fifth of 3x.So remember, what we really want to do is come up with an expression that describes this function, so that we can take itsderivative.Recall that the cosine to the fifth power is equivalent to thecosine raised to the third power times 5. It's now clear that this internal portion of the expression is u raised to the fifth power.Therefore, the composition is something to the fifth power times5, where that something is equal to cosine 3x.The derivative of u is equal to cosine 3x derivative of u. This is the chain rule because we have something on the inside that is notstrictly the letter x. I have a composition: 3x.So take the derivative of the outside, derivative of cosine,that's negative sine, inside stays the same,and then we multiply by the derivative of the inside.So we multiply by the derivative of 3x, which is 3.So this is what goes right here, times negative sine 3x times 3.And then to clean this all up, 5 and a negative times a 3 makes it negative 15, cosine of 3x to the fourth, times the sine of 3x.