`f(x) = x^2 + 5 - 3x^-2` Find the derivative of the function.

Textbook Question

Chapter 2, 2.2 - Problem 39 - Calculus of a Single Variable (10th Edition, Ron Larson).
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kalau's profile pic

kalau | (Level 2) Adjunct Educator

Posted on

The derivative of the first term can be solved using the power rule.

`d/dx x^n = nx^(n-1)`

The derivative of `x^2`  is `2x` .

The derivative of a constant is always zero.

Use the power rule again to solve `-3x^(-2)` .

`-2(-3)(x^(-3)) = 6/x^3`

Combine the terms.  

The derivative is:  `2x+6/x^3`

loves2learn's profile pic

loves2learn | (Level 3) Salutatorian

Posted on

You can also use a Power Rule for the first 2 terms and then use a quotient rule for the last term by bringing the negative exponent down.

Given,

`y=a/b `

Then,

`y'=((a')(b)-(a)(b'))/(b^2) `

Therefore, if we are given,

`f(x)=x^2+5-3x^-2 `

Then, it can be written as,

`f(x)=x^2+5-3/x^2 `

Take the derivative,

`f'(x)=2x+((0)(x^2)-(3)(2x))/x^4 `

Simplify it all down and you should get,

`f'(x)=2x+6/x^3 `

Note that you should get the same answer if you do the other method of using a power rule on all three terms. If you don't, then check your work because you made a mistake along the way.

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