Find the Second Derivative s=6t^5-5t^6

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 $n{t}^{n-1}$ 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, $f^{\prime }(t)$.

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:

1. 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 $f^{\prime }(t)=g^{\prime }(t)+h^{\prime }(t)$.

2. 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 $f^{\prime }(t)=c\cdot g^{\prime }(t)$, where $c$ is a constant.

3. Power Rule: This is a basic rule for differentiation. If $f(t)=t^n$, then the derivative of $f$ with respect to $t$ is $f^{\prime }(t)=n\cdot t^{n-1}$.

4. Derivatives of Polynomial Functions: A polynomial function is composed of terms in the form of $c{t}^n$ 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.