Differentiate:
h(t)=5sin(arccost)
To differentiate the function h(t) = 5sin(arccos(t)), we can use a combination of the chain rule and trigonometric derivatives.
Step 1: Find the derivative of the outer function:
Derivative of sin(u) = cos(u)
Step 2: Find the derivative of the inner function using the chain rule:
Derivative of arccos(t) = -1 / sqrt(1 - t^2)
Step 3: Apply the chain rule:
Derivative of h(t) = (cos(arccos(t))) * (-1 / sqrt(1 - t^2))
Step 4: Simplify the expression using trigonometric identities:
cos(arccos(t)) simplifies to just t
Therefore, the derivative of h(t) with respect to t is:
h'(t) = -t / sqrt(1 - t^2)
To summarize:
The derivative of h(t) = 5sin(arccos(t)) is h'(t) = -t / sqrt(1 - t^2).