find derivative d/dt of sec * square root of t
d/dt ( sec t * t^.5)
=sec t (.5 t^-.5 ) + t^.5 sec t tan t
To find the derivative of the function f(t) = sec(t) * √t with respect to t (denoted as d/dt), we can use the product rule and the chain rule.
The product rule states that if we have two functions u(t) and v(t), then the derivative of their product is given by:
(d/dt)(u(t) * v(t)) = u'(t) * v(t) + u(t) * v'(t)
In this case, our two functions are u(t) = sec(t) and v(t) = √t.
First, let's find the derivative of u(t) = sec(t) using the chain rule. The derivative of sec(t) can be expressed as:
(d/dt)(sec(t)) = sec(t) * tan(t)
Next, let's find the derivative of v(t) = √t. Applying the power rule, we get:
(d/dt)(√t) = (1/2) * t^(-1/2)
Now, we can apply the product rule to find the derivative of the function f(t) = sec(t) * √t. Combining the derivatives we found earlier, we have:
(d/dt)(sec(t) * √t) = (sec(t) * tan(t)) * √t + sec(t) * (1/2) * t^(-1/2)
Simplifying, we get:
(d/dt)(sec(t) * √t) = sec(t) * tan(t) * √t + (1/2) * sec(t) * t^(-1/2)
Therefore, the derivative of the function f(t) = sec(t) * √t with respect to t is:
d/dt(sec(t) * √t) = sec(t) * tan(t) * √t + (1/2) * sec(t) * t^(-1/2)