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Derivative Formulas with Examples, Differentiation Rules

derivative formula

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

 

More from Calculus
Relation and Functions Limits Formula
Continuity Rules Differentiability Rules
Integral Formula Inverse Trigonometric function
Formulas
Application of Integrals Logarithm Formulas

 

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