find the derivative of s(t)=(1+e^t)ln t
To find the derivative of the function s(t) = (1 + e^t)ln(t), we can use the product rule and the chain rule. Here are the steps to find the derivative:
Step 1: Apply the product rule. The product rule states that if you have two functions, u(t) and v(t), then the derivative of their product is given by (u'v + uv').
In this case, let u(t) = (1 + e^t) and v(t) = ln(t).
Step 2: Find the derivatives of u(t) and v(t).
To find the derivative of u(t), we need to use the chain rule. The chain rule states that if you have a composite function, f(g(t)), then the derivative of the composite function is given by (f'(g(t)) * g'(t)).
In this case, f(x) = 1 + x, and g(t) = e^t. The derivative of f(x) with respect to x is 1, and the derivative of g(t) with respect to t is e^t.
So, we have u'(t) = f'(g(t)) * g'(t) = 1 * e^t = e^t.
To find the derivative of v(t), we can use the derivative of ln(t), which is 1/t.
So, v'(t) = 1/t.
Step 3: Apply the product rule.
Using the product rule, the derivative of s(t) becomes:
s'(t) = (u'v + uv') = (e^t * ln(t)) + ((1 + e^t) * (1/t)).
Now, we have the derivative of s(t) as s'(t) = e^t * ln(t) + (1 + e^t) * (1/t).
And that's how you find the derivative of s(t) = (1 + e^t)ln(t).