**– Gradients, derivatives, second derivatives, derivative from first principle**

\(\frac{dy}{dx}\) is the rate of change of y with respect to x

\(\frac{dy}{dx}\) = f'(x) and \(\frac{d^2y}{dx^2}\) = f”(x)

Differentiation from first principle is

\(\frac{dy}{dx}\) = lim \(\frac{f(x +h)-f(x)}{h}\)

Sometimes, h=∆x or δx

**– Derivative of \(x^n\)**

\(\frac{dy}{dx}\) = lim \(\frac{(x +h)^n-(x)^n}{h}\) = nx^{n-1}

**– Gradients, Tangents and Normals ; Maxima and Minima**

For a local maximum or a local minimum f'(x)=0

f ” (x)> 0 implies a minimum

f ”(x)<0 implies a maximum

f ”(x)= 0 is a point of inflection

**– Product rule, Quotient Rule, Chain rule**

Product rule is (u(x)v(x))’ = u'(x)v(x) -u(x)v'(x)

Quotient rule is \(\frac{u(x)}{v(x)}\) ‘ =

**– Functions defined parametrically**

The cycloid is defined by the parametric equations

The slope of the tangent at any point is \(\frac{dy}{dx}\) = \(\frac{sinθ}{1- cosθ}\)

**– Differential equations and applications**

\(\frac{dy}{dx}\) = 3x +2 The solution is y = \(\frac{3}{2}x^2\) + 2x +C

\(\frac{dy}{dx}\) = \(5x^3y^3\) , The solution is y = ± \(\sqrt{\frac{3}{2(5x^3 + C)}}\)