2. Use appropriate differentiation techniques to determine...

2. Use appropriate differentiation techniques to determine the first derivatives of the following functions (Simplify your answer as far as possible) (a) ( y=left(x^{3}+7 x^{2}-x+1 ight)^{4} ) (b) ( y=sqrt{1+2 x-x^{2}} ) (c) ( y=sqrt{frac{1-sqrt{x}}{1+sqrt{x}}} ) (d) ( y=(sin x)^{cos x} ) (e) ( y=e^{sqrt{x}} sqrt{1-x^{2}} )

Transcript

00:01 This is going to take a long time if i try to simplify as much as possible.

00:06 But this first one is just the chain rule where you have to move the 4 in front, now to the third power.

00:12 Leave the inner piece alone, but then you have to multiply by the derivative of the inside which would be your power rule 3x squared plus 14x.

00:24 Notice i'm multiplying the exponent by the coefficient and subtracting 1 from the exponent.

00:30 Because you could simplify this, but to what end and it wouldn't be helpful.

00:34 So i would just leave the answer like this.

00:37 With the next one though, letter b, this is the same thing as to the 1 half power.

00:43 So just like i did the chain rule here, move the 1 half in front, leave the inner piece alone, 1 plus 2x minus x squared, but then multiply by the derivative of the inside which would be 2 minus 2x.

01:01 Now this one actually makes sense to simplify because you can distribute this 1 half in over here and that will give me 1 minus x and the negative exponent will put that into the denominator.

01:17 And then the 1 half power is the square root as i mentioned before.

01:20 So that one actually made sense to simplify.

01:23 Letter c, this one is going to seem complicated, but really what's happening is the entire thing is to the 1 half power.

01:33 So i'm going to bring that 1 half in front to the negative 1 half power, just double checking.

01:39 Leave the inner piece alone, 1 minus root x over 1 plus root x.

01:46 Now take the derivative of the inside which is a quotient rule, derivative of the top would be negative 1 half x to the negative 1 half power.

01:55 Leave the bottom alone, minus, didn't give myself enough room, minus, now the derivative of the bottom, leaving the top alone, all over the denominator squared.

02:18 And i'll admit i don't even want to simplify this because it's going to be gross.

02:24 Like you'd have to distribute, distribute, combine like terms, see if anything happens.

02:31 Even this negative 1 half power is really gross.

02:34 So i'm just going to stop it there, it's going to take me a while.

02:38 With the next one though, what i would do is take the natural log on both sides, natural log of sine of x to the cosine of x power...

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