- #1
gabby989062
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Homework Statement
This is one step of a larger problem, but I'm stuck on derivative of 1/x.
Homework Equations
The Attempt at a Solution
1/x = x^-1.
Using power rule:
-x^-2
but this isn't right?
gabby989062 said:Homework Statement
This is one step of a larger problem, but I'm stuck on derivative of 1/x.
Homework Equations
The Attempt at a Solution
1/x = x^-1.
Using power rule:
-x^-2
but this isn't right?
sponsoredwalk said:[tex] f(x) \ = \ \frac{1}{x} \ = \ \frac{1}{x^1} \ = \ x^{(-1)}[/tex]
[tex] f'(x) \ = \ (-1)x^{(-1) - 1} \ = \ (-1)x^{(-2)} \ = \ \frac{(-1)}{x^{2}} \ =\ - \ \frac{1}{x^2} [/tex]
Proof?
[tex] f(x) \ = \ \frac{1}{x} [/tex]
[tex] f(x \ + \ h) \ = \ \frac{1}{x + h}[/tex]
[tex] f(x \ + \ h) \ - \ f(x) \ = \ \frac{1}{x + h} \ - \ \frac{1}{x} [/tex]
[tex] f(x \ + \ h) \ - \ f(x) \ = \ \frac{1}{x + h} \ - \ \frac{1}{x} \ \Rightarrow \ \frac{x \ - \ x \ - \ h }{x(x + h)} [/tex]
[tex] \lim_{h \to 0} \ \frac{f(x \ + \ h) \ - \ f(x)}{h} \ = \ ? [/tex]
Can you finish it off person who originally asked this question over 2 years ago? :tongue:
The derivative of the function 1/x, where x is a variable, is -1/x². This means that if y = 1/x, then the derivative dy/dx = -1/x².
The derivative of 1/x can be derived using the power rule of differentiation. The function 1/x can be written as x⁻¹. Applying the power rule, we multiply by the exponent (-1) and reduce the exponent by 1, resulting in -1 times x to the power of -2, or -1/x².
Graphically, the derivative of 1/x represents the slope of the tangent to the curve of y = 1/x at any point. It indicates how the function y = 1/x is changing at each point along the curve.
Yes, the derivative of 1/x has practical applications in various fields such as physics, economics, and engineering. It's often used in problems involving rates of change, optimization, and in the analysis of inverse relationships.
The derivative of 1/x, which is -1/x², is defined for all values of x except x = 0. At x = 0, the function 1/x and its derivative both are undefined, as division by zero is not defined in mathematics.
The higher order derivatives of 1/x can be found by repeatedly applying the power rule. For instance, the second derivative of 1/x is 2/x³, and the third derivative is -6/x⁴, and so on. Each differentiation step follows the power rule, multiplying by the current exponent and then decreasing the exponent by one.