Find the derivative. s=t^6tan(t)-sqrt t
If you mean t^6 tan t - √t
just use the product rule on the first term:
ds/dt = 6t^5 tan t + t^6 sec^2 t - 1/(2√t)
That can be massaged in various ways:
t^5(6tan t + t sec^2 t)
t^5(t tan^2 t + 6tan t + t) - 1/2√t
s = t^6 * tan(t) - sqrt(t)
ds/dt = t^6 (sec^2 (t)) + 6t^5 (tan(t)) - (1/2)(1/sqrt(t))
correct
To find the derivative of the given function s = t^6tan(t) - sqrt(t), we will use the rules of differentiation step by step.
Step 1: Apply the product 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)
Let's apply the product rule to our function:
s = t^6 * tan(t) - sqrt(t)
Let u(t) = t^6 and v(t) = tan(t) - sqrt(t).
Then the derivatives are:
u'(t) = 6t^5
v'(t) = sec^2(t) - (1/2)*t^(-1/2)
Step 2: Combine the derivatives using the product rule
(d/dt)(u(t) * v(t)) = u(t) * v'(t) + u'(t) * v(t)
s' = (t^6) * (sec^2(t) - (1/2)*t^(-1/2)) + (6t^5) * (tan(t) - sqrt(t))
Now, simplify the expression further if needed. Keep in mind that trigonometric functions like tan(t), sec^2(t), and sqrt(t) can also be simplified depending on the exact problem or context.
This is how you find the derivative of the function using the product rule and simplify the expression.