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## Chain Rule by decomposition

The Chain Rule applies to any composite function, but we will only deal with composite functions in which the last operation is one of the following:

a trigonometric function (sine, cosine, tangent, etc.), exponent, logarithm, natural logarithm and raise to a power

Any composite function can be written in the form:

y = f(g(x)) = f g(x)

This can be decomposed into y = f(u) where u = g(x).

The Chain Rule says that the derivative of y with respect to the variable x is given by: The steps are:

1. Decompose into outer and inner functions.
2. Differentiate both functions.
3. Multiply the derivatives.
4. Substitute back the original variable.
5. Tidy up.

### Examples

This example may help you to follow the chain rule method. We apply the method to some more examples.

Now lets start with the function:

### Exercises

Differentiate each of these composite functions with respect to x.

1. y = e2x−1
2. y = ln (5x −3)
3. y = (4 − x)7
4. y = sin (x + 4)
5. y = ecos x
6. y = ln (5x3 − 12)
7. y = cos (x4)
8. y = cos4 x
9. y = ln (3x + 1 )
10. y = sin2x         Check all answers (this will open a new window)

A function can be the composition of several functions in succession. We then have to apply the chain rule several times.

### Exercises

Differentiate each equation.

1. 2. 3. 4. 5. Check all answers (this will open a new window)
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