On this problem we have to use chain rule, which is:

So in this problem we let

and

.

Since

and

,

we can conclude that

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 the power rule). Then, we multiply by the derivative of the innermost function:

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REQUIRED KNOWLEDGE

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This problem requires us to understand two things:

1. The derivative of the function is always by itself

2. By adding an operation to the variable in the exponent (in our case, the -s instead of just s), we must multiply the derivative by the derivative of the argument in the exponent. This is an application of the chain rule of derivation

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SOLUTION STEPS

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Thus:

The -1 comes from the derivative of -s

Thus, the correct answer is:

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CORRECT ANSWER

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PROBLEMS?

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If you did not understand the concepts required to solve this derivative problem:

• Look into the derivative of e raised to an argument
• Practice applying the chain rule to derivative problems

We know how to take the derivative of , but not , so let's use the chain rule.

According to the chain rule, we should take the derivative of the outside function and multiply it by the derivative of the inside function. This gives us:

Remember that .

Our final answer:

We start by taking the derivative of both sides of the equation, and proceeding as follows,

.

To find we must use implicit differentiation, which is an application of the chain rule.

Taking of both sides of the equation, we get

using the following rules:

, ,

Note that for every derivative of a function with y, the additional term appears; this is because of the chain rule, where , so to speak, for the function it appears in.

Using algebra to rearrange, we get

For a chain rule derivative, we need to work our way inward from the very outermost function. First, we need to do a power rule for the outer exponent. Then, we multiply that by the derivative of the inside.

Given the function , we can find its derivative using the chain rule, which states that

where and for . We have and , which gives us

We are going to use three rules along with the chain rule:

So then, using our first rule and the chain rule

then using our second rule and chain rule

then using our third rule (no chain rule this time)

Then we rearrange the equation for simplification,

The chain rule of the derivative always deals with the composition of two or more functions.

In this case we can identify two,

and

.

So is the composition of these two such that:

With chain rule, you always start on the outermost function and work your way inward, which in this case is:

Always the derivative of the outermost evaluated at the inner, multiplied by the derivative of the inner.