Question

Please walk through this with me. I need to find the derivative and also the tangent line of each equation

Answer 1

We have;

Power law;

$\frac{d(x^{n)}}{dx}=nx^{n-1}$

$f(x)=\sqrt{x+4}$$f^{\prime}(x)=\frac{1}{2}(x+4)^{-\frac{1}{2}}=\frac{1}{2\sqrt{x+4}}$

ii. At P (0,2);

$f^{\prime}(0)=\frac{1}{2\sqrt{4}}=\frac{1}{4}$

The equation of the tangent line is;

$\begin{gathered} y-2=\frac{1}{4}(x-0) \\ 4(y-2)=x \\ 4y-8=x \\ The\text{ equation of the line is thus }4y-x-8=0 \end{gathered}$

b.

$\begin{gathered} f(x)=x^2+x \\ f^{\prime}(x)=2x+1 \\ f^{\prime}(1)=2(1)+1=3 \end{gathered}$

The equation of the tangent line is;

$\begin{gathered} y-2=3(x-1) \\ y-2=3x-3 \\ The\text{ equation of the line is thus }y-3x+1=0 \end{gathered}$