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Problem 9. Calculate the following = derivatives. Simplify as much as possible: a) Y'(x)= ? y =x sin * ;a"y 6) Find dm if y=sinx...

Question

Problem 9. Calculate the following = derivatives. Simplify as much as possible: a) Y'(x)= ? y =x sin * ;a"y 6) Find dm if y=sinx

Problem 9. Calculate the following = derivatives. Simplify as much as possible: a) Y'(x)= ? y =x sin * ; a"y 6) Find dm if y=sinx



Answers

Find the derivatives of the functions defined as follows.
$$y=\sin \left(\ln 8 x^{6}\right)$$

We're looking for the derivative of Why is equal Teoh Negative X cubed coastline of X plus three X squared sedative X plus six x co. Sign of eggs minus six sign of x To solve for this, we're going to use thes sub difference rule to break down our derivative Be some difference. Rule is F plus minus G prime is equal Teoh f prime plus minus cheaper Applying this rule are derivative Will look like minus D over dx of x cute co sign X plus de over T x three X squared Sign of ass plus D over DX of six co sign of X put minus de over D X of six Sign. Relax. So now we went to start solving for all of these derivatives We've broken down starting with D over DX as ex cute co sign X to solve For this derivative, we would to use the product rule f times G prime is equal to ask time terms Jeep plus f times g prime here are s is equal to x huge So our f prime will be three x squared RG is coastline of X so our G prime will be negative side effects Putting it all together We will get three X squared co sign that X plus negative sign of X Well supplied by X Cute simplified that will look like three X squared co sign of X minus X cubed site of X This is one derivative down. The next derivative will be D over DX of three X squared site of acts to find this derivative were also using the product rule here R f is s squared so our f prime will be to S R G is side of X so I keep time will be co side of X putting use. Together we will get three multiplied by two x sign of X plus co sign of X terms X squared. Our next derivative will be D over DX of six x co side of s. To do this, we use the product rule again. So our f is X last time will be one RG is co side of X So our g prime will be negative side of X Putting these together we will get six times one times coastline X close, negative sign Look x Times X. Now we're going to put all of the derivatives we've taken together, which will be okay. Negative. Three X Squared Co. Sign of X minus X cute side of X plus three times two x sign of X Plus Co sign the X Times X Squared plus six co. Side of X minus six. Sign of X. Let's just take this down. Co spin of X minus six Sign of X minus six co side of X. If we take all of these and we simplify, we will get X huge sign of X.

The function has provided us why it has to access to the Palace in discussion. We need to find that derivative of a given function, which is why all where the X. So first people understand that basic rule which is powerful of the differentiation, supposed access to power and is an expression. So it can be differentiated as and access to power and minus one here and will be six and S. Accessing. So it can be differentiated as B. Bye or V. X equals two six months compared to access to about 6 -1, which equals two six months prepared by access to about five. So this is that required deliberative of the given expression.

Columbus, it's Huang defies the d rodeo. Why equals nineteen Nisei five Max to our promise, Just like your life is constant Thiss part five cy as to the fourth Let fly derivative Offsay Axe Channel cause I tells him

Where this problem we're finding the derivative of negative six extra. The fourth sign. Six sects. For this, we're gonna need to do the product rule. So I have two functions being multiplied together. We have the negative six x to the fourth, and then we have the sign of six X. So for the product rule, we're gonna go ahead and get that D y d x is equal to we take the derivative of the first, which is negative 24 x cubed times The second sign six x plus the first thing negative six x to the fourth times, the derivative of the second, which is the co sign of six X. And then we multiply that whole thing by six because of the chain rule and the derivative of six exes. Six. So then we can go ahead and simplify this thing. So you get negative. 24 X cube signed six x. I'm gonna put the sixes together minus 36 extra. The fourth co sign of six X. You may also see this factored so they have a common factor of negative 12 x cubed. And then what would be left if I kept that is signed to the six X Plus three ex co sign six x So you may also see it like that, depending on your teacher, but they're both correct.


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