The derivative of -1/×
The derivative of -1/× is 1/ײ.
To find the derivative of -1/x, we can use the power rule for differentiation. The power rule states that the derivative of x^n is n*x^(n-1).
In this case, we have -1/x, which can be rewritten as -x^(-1). Using the power rule, we differentiate -x^(-1) as follows:
Differentiating -x^(-1):
-1 * (-1) * x^(-1-1) (Apply the power rule and multiply by the derivative of -1, which is 0)
Simplifying this expression:
-1 * -1 * x^(-2) = x^(-2)
Therefore, the derivative of -1/x is x^(-2).
To find the derivative of -1/x, we can use the power rule of differentiation. The power rule states that the derivative of x^n is equal to n * x^(n-1), where n is a constant.
In this case, we have -1/x, which is equivalent to -x^(-1). Applying the power rule, we can differentiate -1/x as follows:
First, bring the exponent down as a coefficient and decrease the exponent by 1:
-1 * (-1)x^(-1-1)
Simplifying this expression, we get:
1 * x^(-2)
Therefore, the derivative of -1/x is equal to x^(-2).
Alternatively, we can also use the quotient rule to differentiate -1/x. The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative is given by [g'(x) * h(x) - g(x) * h'(x)] / [h(x)]^2.
For -1/x, let g(x) = -1 and h(x) = x. Taking the derivatives of g(x) and h(x), we have g'(x) = 0 and h'(x) = 1.
Applying the quotient rule, we get:
[0 * x - (-1) * 1] / [x]^2
which simplifies to:
1/x^2
Therefore, the derivative of -1/x is x^(-2), which can also be written as 1/x^2.