Differentiation

Implicit Differentiation

When differentiating a function of one variable with respect to another (ie $\frac{dy}{dx}f(y)$), simply differentiate with respect to $y$, then multiply by $\frac{dy}{dx}$.

For example, find $\frac{dy}{dx}$ where $x^2{y}^3-x^2+3y-3=0$. First, using the product rule to differentiate the first term: $$u=x^2{u}^{\prime }=2x$$ $$v=y^3{v}^{\prime }=3y^2\frac{dy}{dx}$$

The equation with all terms differentiated: $$2x{y}^3+3x^2{y}^2\frac{dy}{dx}-2x+3y\frac{dy}{dx}=0$$

Rearranging to get in terms of $\frac{dy}{dx}$: $$3x^2{y}^2\frac{dy}{dx}+3y\frac{dy}{dx}=2x-2x{y}^3$$ $$\frac{dy}{dx}=\frac{2x-2x{y}^3}{3x^2{y}^2+3}$$

Inverse Trig Functions

All the derivatives of the inverse trig functions are given in the data book. They can be derived as follows ($\sin$ is used as an example).

$$y={\sin }^{-1}x$$ $$\sin y=x$$

Differentiating both sides with respect to x

$$\frac{dy}{dx}\cos y=1$$ $$\frac{dy}{dx}=\frac{1}{\cos y}$$

Using pythagorean identity $\cos y=\sqrt{1-{\sin }^{2}y}$

$$\frac{dy}{dx}=\frac{1}{\sqrt{1-{\sin }^{2}y}}=\frac{1}{\sqrt{1-x^2}}$$

Differentials

Differentials describe small changes to values/functions $$y=f(x)$$ $$\frac{dy}{dx}=f^{\prime }(x)$$ $$dy=f^{\prime }(x)dx$$

Recall that $\frac{dy}{dx}\approx \frac{\delta y}{\delta x}$. This means this can be rewritten: $$\delta y\approx f^{\prime }(x)\delta x$$

Dividing both sides by $y=f(x)$: $$\frac{\delta y}{y}=\frac{x{f}^{\prime }(x)}{f(x)}\frac{\delta x}{x}$$

$\frac{\delta y}{y}$ represents a relative change in y, and $\frac{\delta x}{x}$ represents a relative change in x. This can be used to give approximations of how one quantity changes based upon another.

For example, given the mass of a sphere $M=\frac{4}{3}\rho \pi r^3$, where $\rho$ is the material density, estimate the change in mass when the radius is increased by 2%. $$M=\frac{4}{3}\rho \pi r^3$$ $$\frac{dM}{dr}=4\rho \pi r^2$$ $$\delta m=4\rho \pi r^2\delta r$$

Dividing both sides by the original formula: $$\frac{\delta M}{M}=\frac{4\rho \pi r^2\delta r}{\frac{4}{3}\rho \pi r^3}=3\frac{\delta r}{r}$$

$\frac{\delta r}{r}$ represents a relative change in radius, so when $r$ increases by 2%, $\frac{\delta r}{r}=0.02$ $$\frac{\delta M}{M}=3\times 0.02=0.06$$

Meaning the mass increases by 6%.

Hyperbolic Functions

Hyperbolic functions have similar identities to circular trig functions. They're the same, except anywhere there is a product of two $\sinh$s, the term should be negated. Hyperbolic functions can also be defined in terms of exponential functions, making them easy to differentiate.

$$y=\cosh x=\frac{1}{2}(e^x+e^{-x})$$ $$\frac{dy}{dx}=\frac{1}{2}(e^x-e^{-x})=\sinh x$$

All the derivatives of hyperbolic functions are given in the formula book.

Parametric Differentiation

For a function given in parametric form $y=f(t)$, $x=f(t)$: $$\frac{dy}{dx}=\frac{dy}{dt}\times \frac{dt}{dx}$$ $$\frac{d}{dx}=\frac{dt}{dx}\times \frac{d}{dt}$$

$$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{dt}{dx}\times \frac{d}{dt}(\frac{dy}{dx})$$

Partial Differentiation

For a function of two variables $z=f(x,y)$ there are two gradients at the point $z$, one in $x$ and one in $y$. To find the gradient in the x direction, differentiate $f(x,y)$ treating y as a constant. To find the gradient in the y direction, differentiate $f(x,y)$ treating x as a constant. These are the two partial derivatives of the function, $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

For example, for a function $z=4x^3+5y^{7}x$: $$\frac{\partial z}{\partial x}=12x^2+5y^7$$ $$\frac{\partial z}{\partial y}=35y^{6}x$$

Implicit Partial Differentiation

When a function of several variables is given and a partial derivative is required, differentiate the numerator of the partial derivative implicitly with respect to the denominator, and treat the third variable as constant. For example, find $\frac{\partial z}{\partial y}$ given $z^2=x^2+y^2$:

$$\frac{\partial }{\partial y}{z}^2=\frac{\partial }{\partial y}(x^2+y^2)$$

$$2z\frac{\partial z}{\partial y}=2y$$ $$\frac{\partial z}{\partial y}=\frac{y}{z}$$

Another example, find $\frac{\partial z}{\partial x}$ given $z\cos z=x^2{y}^3+z$ $$\frac{\partial }{\partial x}(z\cos z)=\frac{\partial }{\partial x}(x^2{y}^3+z)$$ $$\frac{\partial z}{\partial x}\cos z-z\sin z\frac{\partial z}{\partial x}=2x{y}^3+\frac{\partial z}{\partial x}$$ $$\frac{\partial z}{\partial x}(\cos z-z\sin z-1)=2x{y}^3$$ $$\frac{\partial z}{\partial x}=\frac{2x{y}^3}{\cos z-z\sin z-1}$$

Higher Order Partial Derivatives

Three 2nd order derivatives for functions of 2 variables. For $z=f(x,y)$: $$\frac{{\partial }^{2}z}{\partial x^2}=\frac{\partial }{\partial x}(\frac{\partial z}{\partial x})$$ $$\frac{{\partial }^{2}z}{\partial y^2}=\frac{\partial }{\partial y}(\frac{\partial z}{\partial y})$$ $$\frac{{\partial }^{2}z}{\partial x\partial y}=\frac{\partial }{\partial x}(\frac{\partial z}{\partial y})=\frac{\partial }{\partial y}(\frac{\partial z}{\partial x})=\frac{{\partial }^{2}z}{\partial y\partial x}$$

Note how for the last one, the order is interchangable as it yields the same result.

Chain Rule

The chain rule for a function $w(x,y)$, where x and y are functions of a parameter $t$: $$\frac{dw}{dt}=\frac{\partial w}{\partial x}\frac{dx}{dt}+\frac{\partial w}{\partial y}\frac{dy}{dt}$$

Total Differential

The total differential represents the total height gain or lost when moving along the function described by $z=f(x,y)$ $$dz=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$

Contour Plots

Along a line of a contour plot, the total differential is zero: the height doesn't change. This allows $\frac{dy}{dx}$ to be found $$dh=\frac{\partial h}{\partial x}dx+\frac{\partial h}{\partial y}dy=0$$ $$\frac{dy}{dx}=\frac{\partial h/\partial x}{\partial h/\partial y}$$