Find the Second Derivative s=6t^5-5t^6
You have been presented with a calculus problem which involves differentiation. The specific task here is to determine the second derivative of a given function with respect to variable 't'. The function presented is a polynomial function s(t) = 6t^5 - 5t^6. To find the second derivative, you would first need to differentiate the function to get the first derivative, and then differentiate the first derivative to get the second derivative. This process will involve applying the power rule for differentiation to each term of the polynomial function.
$s = 6 t^{5} - 5 t^{6}$
Answer
Solution:
Step 1: Compute the first derivative of the function.
Step 1.1: Apply the Sum Rule to differentiate $6t^5 - 5t^6$ term by term.
Step 1.2: Differentiate $6t^5$.
Step 1.2.1: The constant 6 is factored out, leaving the derivative of $t^5$.
Step 1.2.2: Apply the Power Rule, which gives $nt^{n-1}$ for $t^n$, to find the derivative of $t^5$.
Step 1.2.3: Calculate the product of 6 and the derivative of $t^5$.
Step 1.3: Differentiate $-5t^6$.
Step 1.3.1: The constant -5 is factored out, leaving the derivative of $t^6$.
Step 1.3.2: Apply the Power Rule to find the derivative of $t^6$.
Step 1.3.3: Calculate the product of -5 and the derivative of $t^6$.
Step 1.4: Combine the results to get the first derivative, $f'(t)$.
Step 2: Compute the second derivative of the function.
Step 2.1: Apply the Sum Rule to differentiate $-30t^5 + 30t^4$ term by term.
Step 2.2: Differentiate $-30t^5$.
Step 2.2.1: The constant -30 is factored out, leaving the derivative of $t^5$.
Step 2.2.2: Apply the Power Rule to find the derivative of $t^5$.
Step 2.2.3: Calculate the product of -30 and the derivative of $t^5$.
Step 2.3: Differentiate $30t^4$.
Step 2.3.1: The constant 30 is factored out, leaving the derivative of $t^4$.
Step 2.3.2: Apply the Power Rule to find the derivative of $t^4$.
Step 2.3.3: Calculate the product of 30 and the derivative of $t^4$.
Step 2.4: Combine the results to get the second derivative, $f''(t)$.
Knowledge Notes:
To solve this problem, we need to understand the following concepts:
Sum Rule: This rule states that the derivative of a sum of functions is the sum of the derivatives of those functions. In mathematical terms, if $f(t) = g(t) + h(t)$, then $f'(t) = g'(t) + h'(t)$.
Constant Multiple Rule: If you have a constant multiplied by a function, the derivative of this is the constant multiplied by the derivative of the function. Mathematically, if $f(t) = c \cdot g(t)$, then $f'(t) = c \cdot g'(t)$, where $c$ is a constant.
Power Rule: This is a basic rule for differentiation. If $f(t) = t^n$, then the derivative of $f$ with respect to $t$ is $f'(t) = n \cdot t^{n-1}$.
Derivatives of Polynomial Functions: A polynomial function is composed of terms in the form of $ct^n$ where $c$ is a constant and $n$ is a non-negative integer. The derivative of a polynomial function is found by applying the Power Rule to each term individually.
By applying these rules, we can find the first and second derivatives of the given polynomial function $s = 6t^5 - 5t^6$. The first derivative is found by differentiating each term separately and then combining the results. The second derivative is found by repeating this process on the first derivative.
