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:
- \(\frac{d}{dx}\)(c) = 0, where c is constant.
- \(\frac{d}{dx}\)(x) = 1
- \(\frac{d}{dx}\)(x^{n}) = n x^{n-1}
- \(\frac{d}{dx}\)[f(x)]^{n} = n [f(x)]^{n-1 }\(\frac{d}{dx}\) f(x)
- \(\frac{d}{{dx}}\sqrt x= \frac{1}{{2\sqrt x }}\)
- \(\frac{d}{dx}\) C∙f(x) = C ∙ \(\frac{d}{dx}\) f(x) = C∙f’(x)
- \(\frac{d}{{dx}}[f(x) \pm g(x)] = \frac{d}{{dx}}f(x) \pm \frac{d}{{dx}}g(x) = f'(x) \pm g'(x)\)
- \(\frac{d}{dx}\) [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. - \(\frac{d}{{dx}}[\frac{{f(x)}}{{g(x)}}] = \frac{{g(x)\frac{d}{{dx}}f(x) – f(x)\frac{d}{{dx}}g(x)}}{{{{[g(x)]}^2}}}\)
This is quotient rule of derivative.
Chain Rule:
- [f(g(x))]’= f’(g(x))g’(x)
- \(\frac{du}{dx}\) = \(\frac{du}{dv}\)∙\(\frac{dv}{dx}\)
- \(\large \frac{du}{dx}=\frac{\frac{du}{dv}}{\frac{dx}{dv}}\)
- \(\large \frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}\)
Logarithm Derivative Formula:
- \(\frac{d}{dx}\)ln x = \(\frac{1}{x}\)
- \(\frac{d}{dx}\)log_{a}x = \(\frac{1}{x\:ln\:a}\)
- \(\frac{d}{{dx}}\ln f(x) = \frac{1}{{f(x)}}\frac{d}{{dx}}f(x)\)
- \(\frac{d}{{dx}}{\log _a}f(x) = \frac{1}{{f(x)\ln a}}\frac{d}{{dx}}f(x)\)
Derivative formula for exponential functions:
- \(\frac{d}{dx}\)e^{x} = e^{x}
- \(\frac{d}{{dx}}{a^x} = {a^x}\ln a\)
- \(\frac{d}{dx}\)e^{f(x)} = e^{f(x)} f’(x)
- \(\frac{d}{dx}\) a^{f(x)} = a^{f(x)}ln a f’(x)
- \(\frac{d}{dx}\) x^{x }= x^{x}(1 + ln x)
Derivative Formula for Trigonometric Formula:
- \(\frac{d}{dx}\) sin x = cos x
- \(\frac{d}{dx}\) cos x = – sinx
- \(\frac{d}{dx}\) tan x = sec^{2} x
- \(\frac{d}{dx}\) cot x = – cosec^{2}x
- \(\frac{d}{dx}\) sec x = sec x∙tan x
- \(\frac{d}{dx}\) cosec x = – cosec x∙cot x
Derivative formula for Inverse Trigonometric functions:
- \(\frac{d}{{dx}}si{n^{ – 1}}x = \frac{1}{{\sqrt {1 – {x^2}} }},{\text{ }} – 1 < x < 1\)
- \(\frac{d}{{dx}}co{s^{ – 1}}x = \frac{{ – 1}}{{\sqrt {1 – {x^2}} }},{\text{ }} – 1 < x < 1\)
- \(\frac{d}{{dx}}ta{n^{ – 1}}x = \frac{1}{{1 + {x^2}}}\)
- \(\frac{d}{{dx}}co{t^{ – 1}}x = \frac{{ – 1}}{{1 + {x^2}}}\)
- \(\frac{d}{{dx}}se{c^{ – 1}}x = \frac{1}{{x\sqrt {{x^2} – 1} }},{\text{ }}\left| x \right| > 1\)
- \(\frac{d}{{dx}}co{\sec ^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {{x^2} – 1} }},{\text{ }}\left| x \right| > 1\)
Derivative formula for hyperbolic functions:
- \(\frac{d}{dx}\) sinh x = cosh x
- \(\frac{d}{dx}\) cosh x = sinh x
- \(\frac{d}{dx}\) tanh x = sech^{2}x
- \(\frac{d}{dx}\) coth x = – cosech^{2}x
- \(\frac{d}{dx}\) sech x = -sech x∙tanh x
- \(\frac{d}{dx}\) cosech x = – cosech x∙coth x
Derivative formula for Inverse Hyperbolic functions:
- \(\frac{d}{{dx}}Sin{h^{ – 1}}x = \frac{1}{{\sqrt {1 + {x^2}} }}\)
- \(\frac{d}{{dx}}Cos{h^{ – 1}}x = \frac{1}{{\sqrt {{x^2} – 1} }}\)
- \(\frac{d}{{dx}}Tan{h^{ – 1}}x = \frac{1}{{1 – {x^2}}},{\text{ }}\left| x \right| < 1\)
- \(\frac{d}{{dx}}Cot{h^{ – 1}}x = \frac{1}{{{x^2} – 1}},{\text{ }}\left| x \right| > 1\)
- \(\frac{d}{{dx}}Sec{h^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {1 – {x^2}} }},{\text{ }}0 < x < 1\)
- \(\frac{d}{{dx}}Co\sec {h^{ – 1}}x = \frac{{ – 1}}{{x\sqrt {1 + {x^2}} }},{\text{ }}x > 0\)
Derivative formula examples:
- Find the derivative of the function given by f(x) = sin (x)^{2}.
Solution:
f(x) = sin(x^{2})
f’(x) = \(\frac{d}{dx}\)( sin x^{2}) x \(\frac{d}{dx}\) x^{2}
= (cos x^{2}) (2x)
= 2x cos x^{2}
- Find \(\frac{dy}{dx}\) if x – y = π.
Solution:
We can write the equation as
y = x – π
\(\frac{dy}{dx}\) = 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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