Question

Hi tutor,Using first principles, find the derivative of y=3x^2+5x-2 please? Here is an image of the formula if you need it:

Answer 1

GIVEN

The function is given to be:

`y=3x^2+5x-2`

SOLUTION

The first principle formula is given to be:

`f^(\prime)(x)=\lim_(h\to0)(f(x+h)-f(x))/(h)`

Rewrite the function:

`f(x)=3x^2+5x-2`

Evaluate f(x + h):

`\begin{gathered} f(x+h)=3(x+h)^2+5(x+h)-2 \\ Expand \\ f(x+h)=3x^2+6xh+3h^2+5x+5h-2 \end{gathered}`

Substitute into the equation:

`f^(\prime)(x)=\lim_(h\to0)(3x^2+6xh+3h^2+5x+5h-2-(3x^2+5x-2))/(h)`

Simplify:

`\begin{gathered} f^(\prime)(x)=\lim_(h\to0)(6xh+3h^2+5h)/(h) \\ \therefore \\ f^(\prime)(x)=\lim_(h\to0)(h(6x+3h+5))/(h) \\ f^(\prime)(x)=\lim_(h\to0)(6x+3h+5) \end{gathered}`

Evaluate the limit:

`f^(\prime)(x)=6x+5`

Therefore, the derivative is:

`f^(\prime)(x)=6x+5`