Explain what is the rule applied to determine f'(x) if f(x)=(x^2+2)^4?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The function f(x)=(x^2+2)^4 can be differentiated using the chain rule. According to the chain rule for f(x) = h(g(x))

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

f(x) = (x^2+2)^4

take g(x) = x^2 + 2, h(x) = x^4

f'(x) = (x^2 + 2)'*4*(x^2 + 2)^3

=> 2x*4*(x^2 + 2)^3

=> 8x*(x^2 + 2)^3

The derivative of the given function is 8x*(x^2 + 2)^3

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The rule applied to determine the first derivative of a composed function is called "chain rule".

f'(x) =[(x^2+2)^4]'*(x^2+2)'

First, we'll differentiate with respect to x,the first factor of the product. We'll differentiate using power rule.

[(x^2+2)^4]' = 4(x^2+2)^3

Now, we'll differentiate with respect to x, the second factor:

(x^2+2)' = 2x + 0

(x^2+2)' = 2x

The first derivative of the given function was determined using chain rule and it is: f'(x) = 8x*(x^2+2)^3.

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