Question

hello please can you help me with the equation find the derivative of the function

Answer 1

You have the following expression:

$f(t)=(5t^2+1)(2\sqrt[]{t}-3t+1)$

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:

$f^{\prime}(t)=(5t^2+1)^{\prime}(2\sqrt[]{t}-3t+1)+(5t^2+1)(2\sqrt[]{t}-3t+1)^{\prime}$

Now, the derivatives of the factors are:

$\begin{gathered} (5t^2+1)^{\prime}=10t \\ (2\sqrt[]{t}-3t+1)^{\prime}=(2t^{\frac{1}{2}}-3t+1)^{\prime}=t^{-\frac{1}{2}}-3 \end{gathered}$

Then, by replacing the previous expressions into f'(t):

$f^{\prime}(t)=10t(2t^{\frac{1}{2}}-3t+1)+(5t^2+1)(t^{-\frac{1}{2}}-3)$

The previous expression is the result for the derivative of f(t).