The chain rule is as follows:

If h(x) = f(g(x)),

h’(x) = g'(x) * f'(g(x))

h(x) is a composite function comprised of one function inside the other; g(x) is the inside function; f(x) is the outside function. g’(x) and f’(x) are the derivatives of the inside and outside function, respectively.

One must determine which function is the inside and outside function. Afterward, the derivative of each function must be determined to apply the chain rule.

It is essential to determine how to determine the outside and inside function.

Hint: Typically (but not all the time), the inside function is in parenthesis or under a square root.

**Example 1 – What is the inside and outside function of h(x) =(x ^{3}+2x)^{2}?**

g(x) = x^{3} + 2x

f(x) = (x)^{2}

**Example 2 – What is the inside and outside function of h(x) = √(5x-8)?**

g(x) = 5x-8

f(x) = √x

**Example 3 – What is the inside and outside function of h(x) = ln(x ^{3} + 2x + 1)?**

g(x) = x^{3} + 2x + 1

f(x) = ln(x)

**Example 4 – What is the inside and outside function of ((2x-3)/(6x+2)) ^{4}?**

g(x) = (2x-3)/(6x+2)

f(x) = (x)^{4}

Now that the inside and outside functions have been established, the derivatives of them must be determined.

**Example 5 – What is the derivative of h(x) =(x ^{3} + 2x)^{2}?**

The inside and outside function has already been determined:

g(x) = x^{3} + 2x

f(x) = (x)^{2}

The derivative of the functions must be determined. Using the power rule for g(x) and f(x):

g'(x) = (3)*x^{3-1} + (1)*2x^{1-1} = 3x^{2} + 2x^{0} = 3x^{2} + 2

f'(x) = (2)*(x)^{2-1} = 2(x)^{1} = 2(x)

Applying the chain rule:

h'(x) = g'(x) * f'(g(x))

h'(x) = (3x^{2} + 2) * 2(x^{3} + 2x)

Simplifying further,

h'(x) = (3x^{2} + 2)*(2x^{3} + 4x) = 6x^{5} + 12x^{3}+ 4x^{3} + 8x = 6x^{5} + 16x^{3} + 8x

**Example 6 – What is the derivative of h(x) = √(5x-8)?**

The inside and outside function has already been determined:

g(x) = 5x-8

f(x) = √(x)

The derivative of the functions must be determined. Using the power rule for g(x) and f(x):

g'(x) = (1)*5x^{1-1} – 0 = 5x^{0} = 5

f'(x) = √(x) = (x)^{1/2} = (1/2)*(x)^{1/2-1} = 1/2*(x)^{-1/2} = 1/(2√(x))

Applying the chain rule:

h'(x) = g'(x) * f'(g(x))

**Example 7 – What is the derivative of h(x) = ln(x ^{3} + 2x + 1)?**

The inside and outside function has already been determined:

g(x) = x^{3} + 2x + 1

f(x) = ln(x)

The derivative of the functions must be determined. Using the power rule for g(x),

g'(x) = (3)*x^{3-1} + (1) * 2x^{1-1} + 0 = 3x^{2} + 2x^{0} = 3x^{2} + 2

f'(x) = 1/(x) (the derivative of ln(x) is 1/x)

Applying the chain rule:

h'(x) = g'(x) * f'(g(x))

**Example 8 – What is the derivative of ?**

The inside and outside function has already been determined:

f(x) = (x)^{4}

The derivative of the functions must be determined. Using the quotient rule for g(x),

In this case, f(x) = 2x-3 and g(x) = 6x+2. Applying the power rule to both functions,

f’(x) = 2

g’(x) = 6

Applying the quotient rule,

Applying the power rule to f’(x),

f'(x) = (4)*(x)^{4-1} = 4(x)^{3}

Applying the chain rule:

h'(x) = g'(x) * f'(g(x))

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