Calculus Course/Differentiation

Derivative
A derivative is a mathematical operation to find the rate of change of a function.

Formula
For a non linear function $$f(x)=y$$. The rate of change of $$f(x)$$ correspond to change of $$x$$ is equal to the ratio of change in $$f(x)$$ over change in $$x$$
 * $$\frac{\Delta f(x)}{\Delta x}=\frac{\Delta y}{\Delta x}$$

Then the Derivative of the function is defined as
 * $$\frac{d}{dx}f(x)=\lim_{\Delta x\to 0}\sum\frac{\Delta f(x)}{\Delta x}=\lim_{\Delta x\to 0}\sum\frac{y}{x}$$

but the derivative must exist uniquely at the point x. Seemingly well-behaved functions might not have derivatives at certain points. As examples, $$f(x)=\frac{1}{x}$$ has no derivative at $$x=0$$ ; $$F(x)=|x|$$ has two possible results at $$x=0$$ (-1 for any value for which $$x<0$$ and 1 for any value for which $$x>0$$) On the other side, a function might have no value at $$x$$ but a derivative of $$x$$, for example $$f(x)=\frac{x}{x}$$ at $$x=0$$. The function is undefined at $$x=0$$, but the derivative is 0 at $$x=0$$ as for any other value of $$x$$.

Practically all rules result, directly or indirectly, from a generalized treatment of the function.

General Rules
$$\frac{d}{dx}(f+g)=\frac{df}{dx}+\frac{dg}{dx}$$

$$\frac{d}{dx}(c\cdot f)=c\cdot\frac{df}{dx}$$

$$\frac{d}{dx}(f\cdot g)=f\cdot\frac{dg}{dx}+g\cdot\frac{df}{dx}$$

$$\frac{d}{dx}\left(\frac{f}{g}\right)=\frac{g\cdot\frac{df}{dx}-f\cdot\frac{dg}{dx}}{g^2}$$

Powers and Polynomials
$$\frac{d}{dx}(c)=0$$

$$\frac{d}{dx}x=1$$

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

$$\frac{d}{dx}\sqrt{x}=\frac{1}{2\sqrt x}$$

$$\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$$

$${\frac{d}{dx}(c_nx^n+c_{n-1}x^{n-1}+c_{n-2}x^{n-2}+\cdots+c_2x^2+c_1x+c_0)=nc_nx^{n-1}+(n-1)c_{n-1}x^{n-2}+(n-2)c_{n-2}x^{n-3}+\cdots+2c_2x+c_1}$$

Trigonometric Functions
$$\frac{d}{dx}\sin(x)=\cos(x)$$

$$\frac{d}{dx}\cos(x)=-\sin(x)$$

$$\frac{d}{dx}\tan(x)=\sec^2(x)$$

$$\frac{d}{dx}\cot(x)=-\csc^2(x)$$

$$\frac{d}{dx}\sec(x)=\sec(x)\tan(x)$$

$$\frac{d}{dx}\csc(x)=-\csc(x)\cot(x)$$

Exponential and Logarithmic Functions
$$\frac{d}{dx}e^x=e^x$$

$$\frac{d}{dx}a^x=a^x\ln(a)\qquad\mbox{if }a>0$$

$$\frac{d}{dx}\ln(x)=\frac{1}{x}$$

$$\frac{d}{dx}\log_a(x)=\frac{1}{\ln(a)x}\qquad\mbox{if }a>0, a\ne 1$$

$$\frac{d}{dx}(f^g)=\frac{d}{dx}\left(e^{g\ln(f)}\right) = f^g\left(\frac{df}{dx}\cdot\frac{g}{f}+\frac{dg}{dx}\cdot\ln(f)\right),\qquad f>0$$

$$\frac{d}{dx}(c^f)=\frac{d}{dx}\left(e^{f\ln(c)}\right)=\frac{df}{dx}\cdot c^f\ln(c)$$

Inverse Trigonometric Functions
$$\frac{d}{dx}\arcsin(x)=\frac{1}{\sqrt{1-x^2}}$$

$$\frac{d}{dx}\arccos(x)=-\frac{1}{\sqrt{1-x^2}}$$

$$\frac{d}{dx}\arctan(x)=\frac{1}{1+x^2}$$

$$\frac{d}{dx}\arcsec(x)=\frac{1}{|x|\sqrt{x^2-1}}$$

$$\frac{d}{dx}\arccot(x)=-\frac{1}{1+x^2}$$

$$\frac{d}{dx}\arccsc(x)=-\frac{1}{|x|\sqrt{x^2-1}}$$

Hyperbolic and Inverse Hyperbolic Functions

 * $$\frac{d}{dx}\sinh(x)=\cosh(x)$$


 * $$\frac{d}{dx}\cosh(x)=\sinh(x)$$


 * $$\frac{d}{dx}\tanh(x)={\rm sech}^2(x)$$


 * $$\frac{d}{dx}{\rm sech}(x)=-\tanh(x){\rm sech}(x)$$


 * $$\frac{d}{dx}\coth(x)=-{\rm csch}^2(x)$$


 * $$\frac{d}{dx}{\rm csch}(x)=-\coth(x){\rm csch}(x)$$


 * $$\frac{d}{dx}{\rm arcsinh}(x)=\frac{1}{\sqrt{x^2+1}}$$


 * $$\frac{d}{dx}{\rm arccosh}(x)=-\frac{1}{\sqrt{x^2-1}}$$


 * $$\frac{d}{dx}{\rm arctanh}(x)=\frac{1}{1-x^2}$$


 * $$\frac{d}{dx}{\rm arcsech}(x)=\frac{1}{x\sqrt{1-x^2}}$$


 * $$\frac{d}{dx}{\rm arccoth}(x)=-\frac{1}{1-x^2}$$


 * $$\frac{d}{dx}{\rm arccsch}(x)=-\frac{1}{|x|\sqrt{1+x^2}}$$

Reference

 * 1) Derivative
 * 2) Table_of_derivatives