Derivative Formula and Differentiation Rules

Question

Learn all Derivative Formula and Differentiation Rules here to clear the concepts.

Answer 1

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 x^n-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) = C∙f’(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. \(\frac{d}{dx}\)log_ax = \(\frac{1}{x\:ln\:a}\)
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. \(\frac{d}{dx}\)e^x = e^x
2. \(\frac{d}{{dx}}{a^x} = {a^x}\ln a\)
3. \(\frac{d}{dx}\)e^f(x) = e^f(x) f’(x)
4. \(\frac{d}{dx}\) a^f(x) = a^f(x)ln a f’(x)
5. \(\frac{d}{dx}\) x^x = x^x(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 x∙tan x
6. \(\frac{d}{dx}\) cosec x = - cosec x∙cot 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 x∙tanh x
6. \(\frac{d}{dx}\) cosech x = - cosech x∙coth 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

2. Find \(\frac{dy}{dx}\) if x – y = π.

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

3. 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