This is a chain rule derivative.  We must first differentiate the natural log function, leaving the inner function as is. Recall:

Now, we must replace this with our function, and multiply that by the derivative of the inner function:

Use chain rule to solve this. First, take the derivative of what is outside of the parenthesis.

You should get .

Next, take the derivative of what is inside the parenthesis. 

You should get .

Multiplying these two together gives .

Using the chain rule, we have

.

Hence, .

Notice that we could have also simplified  first by cancelling the natural log and the exponential function leaving us with just , thereby avoiding the chain rule altogether.

To differentiate the equation above, start by applying the derivative operation to both sides,

Both sides will require the product rule to differentiate,

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 Common Mistake 

A common mistake in the previous step would be to conclude that  instead of  . The former is not correct; if we were looking for the derivative with respect to , then  would in fact be . But we are not differentiating with respect to , we're looking for the derivative with respect to .

We are assuming that  is a function of , so we must apply the chain rule by differentiating with respect to  and multiplying by the derivative of  with respect to  to obtain .

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Collect terms with a derivative onto one side of the equation, factor out the derivative, and divide out  to solve for the derivative , 

Therefore, 

This is a chain rule derivative.  We must first start by taking the derivative of the outermost function.  Here, that is a function raised to the fifth power.  We need to take that derivative (using the power rule).  Then, we multiply by the derivative of the innermost function:

Whenever we have an exponential function with , the first term of our derivative will be that term repeated, without changing anything.  So, the first factor of the derivative will be .  Next, we use chain rule to take the derivative of the exponent.  Its derivative is .  So, the final answer is .

When the function is a constant to the power of a function of x, the first step in chain rule is to rewrite f(x).  So, the first factor of f(x) will be .  Next, we have to take the derivative of the function that is the exponent, or .  Its derivative is 10x-7, so that is the next factor of our derivative.  Last, when a constant is the base of an exponential function, we must always take the natural log of that number in our derivative.  So, our final factor will be .  Thus, the derivative of the entire function will be all these factors multiplied together: .

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get .  The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  To get y' alone, divide both sides by -3 to get .  To simplify even further, we can factor a -2 out of the numerator and denominator and cancel them.  So, the final answer is .

To take the derivative, you must first take the derivative of the outside function, which is sine.  However, the , or the angle of the function, remains the same until we take its derivative later.  The derivative of sinx is cosx, so you the first part of  will be .  Next, take the derivative of the inside function, .  Its derivative is , so by the chain rule, we multiply the derivatives of the inside and outside functions together to get .

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get   The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  To get y' alone, divide both sides by  to get .  To simplify even further, we can factor a 2 out of the numerator and denominator and cancel them.  So, the final answer is .