Even though the coefficients become massive as one progresses through successive derivatives of f(x), the sixth derivative yields a constant. The following seventh derivative will therefore equal zero, as will all following derivatives. We can confidently state — without necessarily deriving the equation ten times — that the tenth derivative will result in an equation equivalent to y=0.

According to the chain rule:

if  , then  .

So, .

To find the second derivative, we will use the chain rule again:

Therefore,   .

Finally, we simply plug in  into the second derivative. 

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: .

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 .

To take the derivative of any exponential function, we repeat the exponential function in the derivative.  So, the first factor of the derivative will be .  Next, we have to take the derivative of the exponent using chain rule.  The derivative of the trigonometric function secx is secxtanx, so in terms of this problem its derivative is .  Since the angle has a scalar of 3, we must also multiply the entire derivative by 3.  So, the answer is .

To find the derivative, we can first rewrite the function to make it easier to take the chain rule.  Rewrite  as .  Now, like in any exponential function, the first factor of the derivative is the original exponential function.  So, the first factor of f'(x) will be .  Next, by the chain rule for derivatives, we must take the derivative of the exponent, which is why we rewrote the exponent in a way that is easier to take the derivative of.  So, the derivative of the exponent is , because the 1/2 and the 2 cancel when we bring the power down front, and the exponent of 1/2 minus 1 becomes negative 1/2.  The last factor of the derivative is  because in every derivative of an exponential function where the base is a number, we must multiply by the natural log of that base.  So, once you multiply all these factors together, the final answer is 

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: .  Next, factor out the y' from both terms on the left side of the equation so that we can solve for it:   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 .

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 and factor out a common y': .  To get y' alone, divide both sides by to get .