Use the function defined below to calculate the indicated derivative.
f(t)= 4/(t^3)+3
what is f''(2)
(second derivative)
Need help!!
Use the power rule,
(xn)' = nxn-1
To find f"(t), repeat the differentiation.
Hint:
4/t³ = 4t-3
Use implicit differentiation to find the slope of the tangent line to the curve
4 x^2 + 2 xy + 1 y^3 = 32
at the point ( -3,2 ).
To calculate the second derivative of a function, you need to differentiate it twice. Let's go step by step.
First, let's find the derivative of f(t).
Given: f(t) = 4/(t^3) + 3
To differentiate f(t), we need to apply the power rule and constant rule:
- The power rule states that if we have a function of the form f(t) = t^n, the derivative is f'(t) = n * t^(n-1).
- The constant rule states that if we have a constant term, its derivative is always zero.
Applying the power rule and constant rule to differentiate f(t), we get:
f'(t) = d/dt[4/(t^3)] + d/dt[3]
Now, let's differentiate each term:
d/dt[4/(t^3)] = (4 * d/dt[(t^(-3))]) = (4 * -3 * t^(-3-1)) = -12/t^4
Note: When differentiating powers of t, we multiply by the power and subtract 1 from the exponent.
d/dt[3] = 0 (since the derivative of a constant is always zero)
Combining the derivatives, we have:
f'(t) = -12/t^4
Now, let's find the second derivative, f''(t), which is the derivative of f'(t):
f''(t) = d/dt[-12/t^4]
To differentiate -12/t^4, we again apply the power rule:
f''(t) = -12 * d/dt[t^(-4)] = -12 * -4 * t^(-4-1) = 48/t^5
So, f''(t) = 48/t^5
Now, to find f''(2), we substitute t = 2 into the second derivative:
f''(2) = 48/(2^5) = 48/32 = 3/2
Therefore, f''(2) is equal to 3/2.
I hope this explanation helps you! Let me know if you have any more questions.