Differentiation I

Bring each power down and reduce it by one. The constant 7 differentiates to 0.

Rewrite as $x^{-2}$, then the power rule gives $-2x^{-3}$.

Chain rule: $7(2x-5)^6$ times the derivative of the inside, which is 2.

Product rule with $u = x$ and $v = \cos x$: $(1)(\cos x) + (x)(-\sin x)$.

At $x=3$, $y = 10$ and $\dfrac{dy}{dx} = 6$. The normal's gradient is $-\tfrac16$, so $y - 10 = -\tfrac16(x-3)$.

$\dfrac{dy}{dx} = 2x - 8 = 0$ gives $x = 4$, and $y = 16 - 32 + 3 = -13$. Since $\dfrac{d^2y}{dx^2} = 2 > 0$, it is a minimum.