Question
Find the derivative of w= z^-5 - 1/z
Answers
GPT 3.5
To find the derivative of w = z^-5 - 1/z, we can use the power rule and the quotient rule.
First, let's find the derivative of z^-5. The power rule states that if we have a function of the form f(x) = x^n, then its derivative is f'(x) = nx^(n-1).
Applying the power rule, we get:
d/dz (z^-5) = -5z^(-5-1) = -5z^-6 = -5/z^6.
Next, let's find the derivative of 1/z. We can rewrite 1/z as z^-1, so we can again use the power rule. Applying the power rule, we get:
d/dz (1/z) = -1z^(-1-1) = -1z^-2 = -1/z^2.
Now, let's find the derivative of w = z^-5 - 1/z using the sum/difference rule. Since the derivative of a sum of functions is the sum of the derivatives of each function, we have:
d/dz (w) = d/dz (z^-5) - d/dz (1/z)
= -5/z^6 - (-1/z^2)
= -5/z^6 + 1/z^2.
Therefore, the derivative of w = z^-5 - 1/z is given by -5/z^6 + 1/z^2.
First, let's find the derivative of z^-5. The power rule states that if we have a function of the form f(x) = x^n, then its derivative is f'(x) = nx^(n-1).
Applying the power rule, we get:
d/dz (z^-5) = -5z^(-5-1) = -5z^-6 = -5/z^6.
Next, let's find the derivative of 1/z. We can rewrite 1/z as z^-1, so we can again use the power rule. Applying the power rule, we get:
d/dz (1/z) = -1z^(-1-1) = -1z^-2 = -1/z^2.
Now, let's find the derivative of w = z^-5 - 1/z using the sum/difference rule. Since the derivative of a sum of functions is the sum of the derivatives of each function, we have:
d/dz (w) = d/dz (z^-5) - d/dz (1/z)
= -5/z^6 - (-1/z^2)
= -5/z^6 + 1/z^2.
Therefore, the derivative of w = z^-5 - 1/z is given by -5/z^6 + 1/z^2.