Find the derivatives of the function g(t)= cosh^3 t.
g(t) = (cosh(t) )^3
g ' (t) = 3 cosh(t) * (-sinh(t))
To find the derivatives of the function g(t) = cosh^3(t), we can use the chain rule. Let's calculate the derivatives step by step:
Step 1: Recall the derivative of cosh t
The derivative of cosh t with respect to t is sinh t. So, we have:
d(cosh t) / dt = sinh t
Step 2: Apply the chain rule
To find the derivative of g(t) = cosh^3(t), we will use the chain rule. Let's call g(t) = u^3 and u = cosh t. Then, we have:
d(g(t)) / dt = d(u^3) / dt
Using the chain rule, we can write this as:
d(g(t)) / dt = d(u^3) / du * du / dt
Step 3: Find d(u^3) / du
To find the derivative of u^3 with respect to u, we can simply apply the power rule for differentiation. The power rule states that if f(u) = u^n, then:
d(f(u)) / du = n * u^(n-1)
So, applying this rule to u^3, we have:
d(u^3) / du = 3 * u^(3-1) = 3 * u^2
Step 4: Find du / dt
To find the derivative of u (which is cosh t) with respect to t, we recall that the derivative of cosh t is sinh t:
du / dt = sinh t
Step 5: Substitute the derivatives into the chain rule expression
Now, we can substitute the derivatives we found in Step 3 and Step 4 into the chain rule expression:
d(g(t)) / dt = (d(u^3) / du) * (du / dt)
= 3 * u^2 * sinh t
Step 6: Substitute u back in terms of t
Now, we plug u back into terms of t, which is u = cosh t:
d(g(t)) / dt = 3 * (cosh t)^2 * sinh t
So, the derivative of g(t) = cosh^3(t) with respect to t is 3 * (cosh t)^2 * sinh t.
To find the derivative of the function g(t) = cosh^3 t, we can use the chain rule. The chain rule states that if we have a composition of functions, where we have a function of a function, then the derivative can be calculated by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
Let's break down the function g(t) = cosh^3 t into its individual components:
g(t) = (cosh t)^3
The outer function in this case is f(x) = x^3, and the inner function is h(t) = cosh t, which is the hyperbolic cosine function.
Now, let's find the derivatives of the outer and inner functions separately:
Taking the derivative of the outer function:
f'(x) = 3x^2
Taking the derivative of the inner function:
h'(t) = sinh t
Now, bringing it all together using the chain rule:
g'(t) = f'(h(t)) * h'(t)
Substituting the derivatives we found earlier:
g'(t) = 3(cosh t)^2 * sinh t
Therefore, the derivative of the function g(t) = cosh^3 t is g'(t) = 3(cosh t)^2 * sinh t.