# Derivative Formula:

Actual derivative formula:
If y = f(x), then
$$\LARGE f’(x)=\lim_{\triangle x \rightarrow 0}\frac{f(x+ \triangle x)-f(x)}{\triangle x}$$

## Basic Derivative formula:

1. $$\frac{d}{dx}$$(c) = 0, where c is constant.
2. $$\frac{d}{dx}$$(x) = 1
3. $$\frac{d}{dx}$$(xn) = n xn-1
4. $$\frac{d}{dx}$$[f(x)]n = n [f(x)]n-1 $$\frac{d}{dx}$$ f(x)
5. $$\frac{d}{{dx}}\sqrt x= \frac{1}{{2\sqrt x }}$$
6. $$\frac{d}{dx}$$ C∙f(x) = C ∙ $$\frac{d}{dx}$$ f(x) = Cf’(x)
7. $$\frac{d}{{dx}}[f(x) \pm g(x)] = \frac{d}{{dx}}f(x) \pm \frac{d}{{dx}}g(x) = f'(x) \pm g'(x)$$
8. $$\frac{d}{dx}$$ [f(x) g(x)] = f(x) $$\frac{d}{dx}$$ g(x) + g(x) $$\frac{d}{dx}$$ f(x)
This is called product rule of derivative.
9. $$\frac{d}{{dx}}[\frac{{f(x)}}{{g(x)}}] = \frac{{g(x)\frac{d}{{dx}}f(x) – f(x)\frac{d}{{dx}}g(x)}}{{{{[g(x)]}^2}}}$$
This is quotient rule of derivative.

## Chain Rule:

1. [f(g(x))]’= f’(g(x))g’(x)
2. $$\frac{du}{dx}$$ = $$\frac{du}{dv}$$$$\frac{dv}{dx}$$
3. $$\large \frac{du}{dx}=\frac{\frac{du}{dv}}{\frac{dx}{dv}}$$
4. $$\large \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}$$

## Logarithm Derivative Formula:

1. $$\frac{d}{dx}$$ln x = $$\frac{1}{x}$$
2. $$\frac{d}{dx}$$logax = $$\frac{1}{x\:ln\:a}$$
3. $$\frac{d}{{dx}}\ln f(x) = \frac{1}{{f(x)}}\frac{d}{{dx}}f(x)$$
4. $$\frac{d}{{dx}}{\log _a}f(x) = \frac{1}{{f(x)\ln a}}\frac{d}{{dx}}f(x)$$

## Derivative formula for exponential functions:

1. $$\frac{d}{dx}$$ex = ex
2. $$\frac{d}{{dx}}{a^x} = {a^x}\ln a$$
3. $$\frac{d}{dx}$$ef(x) = ef(x) f’(x)
4. $$\frac{d}{dx}$$ af(x) = af(x)ln a f’(x)
5. $$\frac{d}{dx}$$ xx = xx(1 + ln x)

## Derivative Formula for Trigonometric Formula:

1. $$\frac{d}{dx}$$ sin x = cos x
2. $$\frac{d}{dx}$$ cos x = – sinx
3. $$\frac{d}{dx}$$ tan x = sec2 x
4. $$\frac{d}{dx}$$ cot x = – cosec2x
5. $$\frac{d}{dx}$$ sec x = sec xtan x
6. $$\frac{d}{dx}$$ cosec x = – cosec xcot x

## Derivative formula for Inverse Trigonometric functions:

1. $$\frac{d}{{dx}}si{n^{ – 1}}x = \frac{1}{{\sqrt {1 – {x^2}} }},{\text{ }} – 1 < x < 1$$
2. $$\frac{d}{{dx}}co{s^{ – 1}}x = \frac{{ – 1}}{{\sqrt {1 – {x^2}} }},{\text{ }} – 1 < x < 1$$
3. $$\frac{d}{{dx}}ta{n^{ – 1}}x = \frac{1}{{1 + {x^2}}}$$
4. $$\frac{d}{{dx}}co{t^{ – 1}}x = \frac{{ – 1}}{{1 + {x^2}}}$$
5. $$\frac{d}{{dx}}se{c^{ – 1}}x = \frac{1}{{x\sqrt {{x^2} – 1} }},{\text{ }}\left| x \right| > 1$$
6. $$\frac{d}{{dx}}co{\sec ^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {{x^2} – 1} }},{\text{ }}\left| x \right| > 1$$

## Derivative formula for hyperbolic functions:

1. $$\frac{d}{dx}$$ sinh x = cosh x
2. $$\frac{d}{dx}$$ cosh x = sinh x
3. $$\frac{d}{dx}$$ tanh x = sech2x
4. $$\frac{d}{dx}$$ coth x = – cosech2x
5. $$\frac{d}{dx}$$ sech x = -sech xtanh x
6. $$\frac{d}{dx}$$ cosech x = – cosech xcoth x

## Derivative formula for Inverse Hyperbolic functions:

1. $$\frac{d}{{dx}}Sin{h^{ – 1}}x = \frac{1}{{\sqrt {1 + {x^2}} }}$$
2. $$\frac{d}{{dx}}Cos{h^{ – 1}}x = \frac{1}{{\sqrt {{x^2} – 1} }}$$
3. $$\frac{d}{{dx}}Tan{h^{ – 1}}x = \frac{1}{{1 – {x^2}}},{\text{ }}\left| x \right| < 1$$
4. $$\frac{d}{{dx}}Cot{h^{ – 1}}x = \frac{1}{{{x^2} – 1}},{\text{ }}\left| x \right| > 1$$
5. $$\frac{d}{{dx}}Sec{h^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {1 – {x^2}} }},{\text{ }}0 < x < 1$$
6. $$\frac{d}{{dx}}Co\sec {h^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {1 + {x^2}} }},{\text{ }}x > 0$$

## Derivative formula examples:

1. Find the derivative of the function given by f(x) = sin (x)2.

Solution:
f(x) = sin(x2)
f’(x) = $$\frac{d}{dx}$$( sin x2) x $$\frac{d}{dx}$$ x2
= (cos x2) (2x)
= 2x cos x2

1. Find $$\frac{dy}{dx}$$ if x – y = π.

Solution:
We can write the equation as
y = x – π
$$\frac{dy}{dx}$$ = 1

1. Find $$\frac{dy}{dx}$$, if y + sin y = cos x.

Solution:
We differentiate the relationship directly with respect to x,
$$\frac{dy}{dx}$$ + $$\frac{d}{dx}$$(sin y) = $$\frac{d}{dx}$$(cos x)
which implies using chain rule
$$\frac{dy}{dx}$$ + cos y $$\frac{dy}{dx}$$ = -sin x

This allows $$\frac{dy}{dx}$$ = – $$\frac{sin x}{1 + cos y}$$
where y ≠ (2n + 1) π