You have the following expression:
[tex]f(t)=(5t^2+1)(2\sqrt[]{t}-3t+1)[/tex]
In order to derivate the previous function, notice that you have a product between two factors, then, by using the derivative of a product you obtain:
[tex]f^{\prime}(t)=(5t^2+1)^{\prime}(2\sqrt[]{t}-3t+1)+(5t^2+1)(2\sqrt[]{t}-3t+1)^{\prime}[/tex]
Now, the derivatives of the factors are:
[tex]\begin{gathered} (5t^2+1)^{\prime}=10t \\ (2\sqrt[]{t}-3t+1)^{\prime}=(2t^{\frac{1}{2}}-3t+1)^{\prime}=t^{-\frac{1}{2}}-3 \end{gathered}[/tex]
Then, by replacing the previous expressions into f'(t):
[tex]f^{\prime}(t)=10t(2t^{\frac{1}{2}}-3t+1)+(5t^2+1)(t^{-\frac{1}{2}}-3)[/tex]
The previous expression is the result for the derivative of f(t).
