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Chain Rule

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)) = fofg(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:

dy by dx is equal to dy by du multiplied by du by dx.

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.

1. Write  the inner function, u, as a function, g, of x. 2. Write the outer function, f, as a function of u. 3. Compute the derivatives of these two functions, f and g. 4. Multiply the two derivatives. 5. Substitute g(x) for u in the product. Simplify if possible.

We apply the method to some more examples.

y equals sine of (x cubed)
y = sin u and u = x cubed
y = sin u and u = x cubed
dy by du equals cos u and du by dx equals 3(x squared)
dy by dx equals 3 times (x squared) times the cosine of (x cubed)

Now lets start with the function:

y equals the (cosine of x) squared
y equals u squared and u equals the cosine of x
y = sin u and u = x cubed
dy by du equals 2u and du by dx equals negative sine of x
dy by dx equals −2 times the (cosine of x) times the (sine of x)

 

More Chain Rule Examples

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.

Example

y equals the sine of the exponent of (x squared)
y equals the sine of u and u equals the exponent of (x squared)
u equals the exponent of v and v = x squared
dy by du equals the cosine of u, du by dv equals the exponent of v and dv by dx equals 2x
dy by dx equals dy by du times du by dv times dv by dx equals 2x times  the exponent of v times the cosine of u
dy by dx equals 2x times (the exponent of (x squared)) times (the cosine of the exponent of (x squared))

More Multiple Chain Rule Examples

Exercises

Differentiate each equation.

  1. y equals
  2. 1 over the cube root of the sine of (5x minus 2)
  3. y equals the sine of (negative 3 times x squared) all raised to the power 4
  4. y equals the exponent of (4 times (x to the power of 5) minus 7)to the power of negative 2
  5. y equals the natural logarithm of the cosine of (4 time (x squared) minus 3)      Check all answers (this will open a new window)

<< Inner and Outer Functions | Chain Rule Index | Concise method >>

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