Differentiate: 1/s+ke^s
Do I just use the quotient rule? Thanks
To differentiate the expression 1/s + ke^s, you can indeed use the quotient rule. However, in this case, the expression can be simplified before applying the rule.
First, let's rewrite the expression using common denominators:
1/s + ke^s = (1/s)(s/s) + ke^s = (1 + s^2)/s + ke^s
Now, we can differentiate this expression using the quotient rule. The quotient rule states that for a function f(x) = g(x)/h(x), where g(x) and h(x) are differentiable functions, the derivative is given by:
f'(x) = (g'(x) * h(x) - g(x) * h'(x)) / (h(x))^2
In our case, let's define:
g(x) = 1 + s^2
h(x) = s
Now, let's calculate the derivatives:
g'(x) = 0 + 2s = 2s
h'(x) = 1
Applying the quotient rule:
f'(x) = (2s * s - (1 + s^2) * 1) / s^2
Simplifying:
f'(x) = (2s^2 - (1 + s^2)) / s^2
Combining like terms:
f'(x) = (2s^2 - 1 - s^2) / s^2
= (s^2 - 1) / s^2
Therefore, the derivative of the expression 1/s + ke^s is:
f'(x) = (s^2 - 1) / s^2