In each part compute f^', f^'', f^''' and then state the formula for f^(n) (a) f(x)=1 / x (b) f(

step 1 So we want to compute the first derivative, second derivative, and third derivative, and state a

In each part compute f^', f^'', f^''' and then state the...

In each part compute $f^{\prime}, f^{\prime \prime}, f^{\prime \prime \prime}$ and then state the formula for $f^{(n)}$
(a) $f(x)=1 / x$
(b) $f(x)=1 / x^{2}$
[Hint: The expression $(-1)^{n}$ has a value of 1 if $n$ is even and $-1$ if $n$ is odd. Use this expression in your answer.]

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Key Concepts

Power Rule

The power rule is a basic differentiation technique stating that if f(x) = x^n, then the derivative f?(x) = n·x^(n?1). This rule is widely used because many functions, including reciprocals expressed as negative powers, can be differentiated easily by rewriting them in the form x^m.

Expressing Rational Functions as Power Functions

Functions given as reciprocals, such as f(x) = 1/x or f(x) = 1/x², can be rewritten using negative exponents (x^(?1) and x^(?2), respectively). This conversion allows the straightforward application of the power rule and facilitates the process of finding higher order derivatives.

nth Derivative Formulas

For functions that are power functions, the nth derivative can be systematically computed by applying the power rule repeatedly. The resulting formulas involve products that decrement the exponent by one at each differentiation, often expressible in terms of factorial expressions, and may include alternating signs when the exponent is negative.

Sign Alternation Using (?1)^n

When differentiating functions expressed as negative powers, the process can yield alternating signs in successive derivatives. Expressing this alternation using (?1)^n provides a concise way to capture the overall sign behavior in the formula for the nth derivative.

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