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Chain Rule

Chain Rule by a concise method

In using the chain rule by the decomposition method we substitute the inner function into the derivative of the outer one at the end. However with a bit of practice, one usually makes the substitution right after each differentiation. That is,

  • Differentiate the outer function
  • Substitute the inner function into the derivative
  • Repeat the above with the inner function if it is composite and multiply them together.

Remember that the Chain Rule can be used on any composite function. A composite function can be thought of as a chain of simpler one operation functions. This function is fofgofhofk(x). Starting with x as the clasp, the operations (links) are added one by one, starting with the innermost function, k, and ending with the outermost function, f.

a

To differentiate the composite function we differentiate the simpler functions (links) one by one, starting with the outermost function, f. Each differentiated piece of chain is multiplied to get the derivative of the full composite function.

a product of the derivatives of each function making up the function

This means that you can compute the derivative of a composite function quickly in your head, so long as you can deduce the order of the functions involved in the composition.

Example

Find the derivative of y = sin(ln(5x2 − 2x)).

We need to differentiate the functions one by one from outermost to innermost. Initially we write down the steps required to make the function, in reverse order, until we get back to an expression that can be differentiated by the basic rules. We can then differentiate:

a

We usually think about what the order of functions is and then deal with them in reverse order.

For derivatives where the Chain Rule is applied once, you should now be practicing simply writing down the complete derivative and tidying the expression.

Examples

Differentiate y = sin(2x). This is a composite function. Sine is the outer function, and double is the inner function. Hence dy by dx = (cos(2x))(2) = 2cos(2x).

Differentiate a

Exercises

Use the concise method to obtain the derivative of each expression:

  1. f(x) = e3x
  2. f(x) = sin(x4)
  3. y = cos(4x3)
  4. y = ln(3x)
  5. f(x) = (1 + 2x)3
  6. f(x) = sin(2x5)
  7. y = cos(x−3)
  8. y = ln(x3 + x)
  9. f(x) = ex + 3
  10. f(x) = sin(x9 − 2)
  11. y = (4 + cos x)3
  12. y = ln(6x2)         Check all answers (this will open a new window)

The next exercises require the application of the chain rule twice. Use the concise method to solve them.

Exercise

Find the value of f '(x) for each of these functions:

  1. f(x) = esin 3x
  2. f(x) = 1 over ( 3 x squared plus 2x plus 1 )
  3. f(x) = ln 4x3
  4. f(x) = ln (sin 3x)
  5. f(x) = square root of the sine of 5x
  6. f(x) = cos (4x0.5 − 3)
  7. f(x) = ecos (6x − 1)
  8. f(x) = sin5 (x5)
  9. f(x) = 4 over the natural log of 3x
  10. f(x) = cosine of the square root of 9x        Check all answers (this will open a new window)

Exercise

Find the derivative of

f(x) =

Working:

f '(x) =

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