Question

differentiate $f(t)=\frac{1}{2}t^{6}-8t^{4}+t$. answer: $f(t)=$

Explanation:

Step1: Apply power - rule to $\frac{1}{2}t^{6}$

The power - rule states that if $y = ax^{n}$, then $y^\prime=anx^{n - 1}$. For $y=\frac{1}{2}t^{6}$, $a = \frac{1}{2}$ and $n = 6$. So the derivative is $\frac{1}{2}\times6t^{6 - 1}=3t^{5}$.

Step2: Apply power - rule to $-8t^{4}$

For $y=-8t^{4}$, $a=-8$ and $n = 4$. So the derivative is $-8\times4t^{4 - 1}=-32t^{3}$.

Step3: Apply power - rule to $t$

For $y = t=t^{1}$, $a = 1$ and $n = 1$. So the derivative is $1\times1t^{1 - 1}=1$.

Step4: Combine the derivatives

Since the derivative of a sum/difference of functions is the sum/difference of their derivatives, $f^\prime(t)=3t^{5}-32t^{3}+1$.

Answer:

$3t^{5}-32t^{3}+1$