Techniques for differentiation

This article contains a list of techniques for the differentiation of real functions, categorized by type.

imple polynomial functions

Given a polynomial $p(x)$, that is defined by the formula:

:$p(x)\; =\; sum^m\_\{i=0\}\; k\_i\; x^i$, one has

:$frac\{d\}\{dx\}\; p(x)\; =\; sum^m\_\{i=0\}\; ik\_ix^\{i-1\}.$

That is, one simply multiplied each term by its degree, then divides by ‘’x’’. For example, one can differentiate $sqrt\{x\}\; +\; 5x$. First, one would break it down into its component terms: sqrt(x) and 5x. sqrt(x) is equal to x^1/2, meaning that its derivative is 1/(2sqrt(x)), or half the reciprocal of the value. 5x simply becomes 5, giving us:

:$frac\{d\}\{dx\}\; (sqrt\{x\}\; +\; 5x)\; =\; frac\{1\; +\; 10sqrt\{x\{2sqrt\{x.$

Exponential functions

Given some function ‘’f(x)’’ equals b^x, its derivative can be found via the following formula:

:$frac\{d\}\{dx\}\; b^x\; =\; b^x\; ln\; b$

where ‘’ln b’’ is the natural logarithm of b. Using this formula, we can differentiate 225^x, which gives us 2(ln 3 + ln 5). (See Natural logarithm). So ultimately, we have b^x 2ln 3 + b^x 2ln 5.

Proof

:$frac\{d\}\{dx\}b^x$

:$=frac\{d\}\{dx\}e^\{ln\; b^x\}$ A property of logarithms.

:$=frac\{d\}\{dx\}e^\{xln\; b\}$ Another property of logarithms

:$=left(frac\{d\}\{dx\}xln\; b\; ight)left(e^\{xln\; b\}\; ight)$ From the chain rule.

:$=left(ln\; b\; ight)left(e^\{ln\; b^x\}\; ight)$

:$=b^x\; ln\; b$

Logarithmic functions

All logarithmic functions can be differentiated via a formula very similar to that for exponential functions. The slope of any logarithmic function at a point "x" is equal to the reciprocal of x times the natural logarithm of the base, or:

:$frac\{d\}\{dx\}\; log\_b\; x\; =\; frac\{1\}\{x\; ln\; b\}.$

Through this we can differentiate the natural logarithm itself. Of course, the base of the natural logarithm is "e", and the base "x" logarithm of "x" is always one. Therefore, the natural logarithm of "e" is one. Knowing this, we can find that the slope of the natural logarithm at any point equals the reciprocal of the height at that point.

Proof

Let $y=log\_b\; x$.

Then $b^y=x$.

:$e^\{ln\; b^y\}=x$

:$e^\{yln\; b\}=x$

Use implicit differentiation.

:$left(e^\{yln\; b\}\; ight)left(frac\{d\}\{dx\}yln\; b\; ight)=1$

:$left(e^\{ln\; b^y\}\; ight)left(frac\{dy\}\{dx\}\; ight)left(ln\; b\; ight)=1$

:$frac\{dy\}\{dx\}=frac\{1\}\{(b^y)(ln\; b)\}$

Since $y=log\_b\; x$ and $b^y=x$, $frac\{d\}\{dx\}log\_b\; x=frac\{1\}\{xln\; b\}$.

imple Trigonometric functions

$frac\{d\}\{dx\}sin\; x=cos\; x$

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

$frac\{d\}\{dx\}\; an\; x=sec^2\; x$$frac\{d\}\{dx\}sec\; x=\; an\; x\; sec\; x$

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

$frac\{d\}\{dx\}cot\; x=-csc^2\; x$col-end

For an extensive list of derivatives of trigonometric functions, hyperbolic functions, their inverses, and proofs, see table of derivatives and Differentiation of trigonometric functions.

ee also

*Calculus
*Derivative
*List of Differentiation Identities