Derivative Formula

Derivative Formulas

Derivative is a rate of change of function with respect to a variable.After the invention of a derivative of a function by Newton and Leibnitz in around 17th century, it is widely used in the sector of math and physics.

Some of the important formulas of derivative are as follows:-

Let u, v, and w are functions of the variable x, and a, b, c are constant then
$1 . \dfrac{d}{dx}(c ) = o \\[3mm] 2. \dfrac{d}{dx}(x) = 1 \\[3mm] 3. \dfrac{d}{dx} (x^n) = nx^{n- 1} \\[3mm] 4. \dfrac{d}{dx}(u \pm v) = \dfrac{du}{dx} \pm \dfrac{dv}{dx} \\[3mm] 5. \dfrac{d}{dx}(cu) = c\dfrac{du}{dx} \\[3mm] 6. \dfrac{d}{dx}(uv) = u \dfrac{dv}{dx} + v \dfrac{du}{dx} \\[3mm] 7. \dfrac{d}{dx}\left(\dfrac{u}{v} \right) = \dfrac{v \dfrac{du}{dx}- u \dfrac{dv}{dx}}{v^2}(\text{quotient rule}) \\[3mm] 8. \dfrac{d}{dx}(u^n) = nu^{n- 1} \dfrac{du}{dx} \\[3mm] 9. \dfrac{d}{dx}(yu, v) = \dfrac{dv}{dx} \left(\dfrac{du}{dx}, y \right) \\ 10. \dfrac{d}{dx} (u, v) = \dfrac{dv}{dx} \left(\dfrac{du}{dx}, v \right) (\text{Chain Rule}) \dfrac{du}{dx} = \dfrac{du}{dv} \dfrac{dv}{dx} \\ \text{Another similar formula is given by} \dfrac{du}{dx} = \dfrac{\dfrac{du}{dv}}{\dfrac{dx}{dv}}$
Derivative of Trigonometric function and their inverse:
$1. \dfrac{d}{dx}(sin(u)) = \cos(u) \dfrac{du}{dx} \\[3mm] 2. \dfrac{d}{dx}(\cos(u)) = -\sin(u) \dfrac{du}{dx} \\[3mm] 3. \dfrac{d}{dx}(\tan(u)) = \sec^2(u) \dfrac{du}{dx} \\[3mm] 4. \dfrac{d}{dx} (\cot(u)) = -\csc^2(u) \dfrac{du}{dx} \\[3mm] 5. \dfrac{d}{dx}(\sec(u)) = \sec(u) \tan(u) \dfrac{du}{dx} \\[3mm] 6. \dfrac{d}{dx}(\csc(u)) = -\csc(u) \cot(u) \dfrac{du}{dx} \\[3mm] 7. \dfrac{d}{dx}(\sin^{-1}(u)) = \dfrac{1}{\sqrt{1- u^2}} \dfrac{du}{dx} \\[3mm] 8. \dfrac{d}{dx}(\cos^{-1}(u)) = -\dfrac{1}{\sqrt{1- u^2}}\dfrac{du}{dx} \\[3mm] 9. \dfrac{d}{dx} (\tan^{-1}(u)) = \dfrac{1}{1 + u^2} \dfrac{du}{dx} \\[3mm] 10. \dfrac{d}{dx}(\cot^{-1}(u)) = -\dfrac{1}{1 + u^2} \dfrac{du}{dx} \\ 11. \dfrac{d}{dx}(\sec^{-1}(u)) = \dfrac{1}{|u|\sqrt{u^2- 1}}\dfrac{du}{dx} \\ 12. \dfrac{d}{dx}(\csc^{-1}(u)) = \dfrac{1}{-|u|\sqrt{u^2- 1}}\dfrac{du}{dx}$
Derivative of the Exponential and Logarithmic
$1. \dfrac{d}{dx}(\ln(u)) = \dfrac{1}{u}\dfrac{du}{dx} \\[3mm] 2. \dfrac{d}{dx}(\log_a(u)) = \dfrac{1}{\ln(a) u} \dfrac{du}{dx} \\[3mm] 3. \dfrac{d}{dx}(e^u) = e^u \dfrac{du}{dx} \\[3mm] 4. \dfrac{d}{dx}(a^u) = \ln(a)a^u \dfrac{du}{dx} \\ 5. \dfrac{d}{dx} (u^v) = u^v . \left[ \dfrac{u'v + uv' \ln u}{u}\right]$
Deviate of the Hyperbolic function and their Inverses
Recall the definition of the trigonometric functions of the trigonometric functions.
$\cos h(x) =\dfrac{e^x + e^{-x}}{2}\\[3mm] \sin h(x) = \dfrac{e^x- e^{-x}}{2} \\[3mm] \tan h(x) = \dfrac{\sin h(x)}{\cos h(x)} \\ \cot h(x) = \dfrac{\cos h(x)}{\sin h(x)} \\ \sec h(x) = \dfrac{1}{\cos h(x)} \\ \csc h(x) = \dfrac{1}{\sin h(x)} \\[3mm] 1. \dfrac{d}{dx}(\sin h(u)) = \cos h(u) \dfrac{du}{dx} \\ 2. \dfrac{d}{dx}(\cos h(u)) = \sin h(u) \dfrac{du}{dx} \\[3mm] 3. \dfrac{d}{dx}(\tan h(u)) = \sec h^2 (u) \\[3mm] 4. \dfrac{d}{dx}(\cot h(u)) = -\csc h^2(u) \dfrac{du}{dx} \\ 5. \dfrac{d}{dx}(\sec h(u)) = -\sec h(u) \tan h(y) \dfrac{du}{dx} \\ 6. \dfrac{d}{dx}(\csc h(u)) = - \csc h(u) \cot h(y)\dfrac{du}{dx} \\ 7. \dfrac{d}{dx}(\sin h^{-1}(u)) = \dfrac{1}{\sqrt{u^2 + 1}} \dfrac{du}{dx}$
$8. \dfrac{d}{dx}(\cos h^{-1}(u)) = \pm \dfrac{1}{\sqrt{y^2- 1}} \dfrac{du}{dx} \\[3mm] 9. \dfrac{d}{dx}(\tan h^{-1}(u)) = \pm \dfrac{1}{\sqrt{u^2- 1}}\dfrac{du}{dx} (-1 < u <1) \\[3mm] 10. \dfrac{d}{dx}(\cot h^{-1}(u)) = \dfrac{1}{1- u^2}\dfrac{du}{dx} (u >1 \,or u< -1) \\[3mm] 11. \dfrac{d}{dx}(\sec h^{-1}(u)) = \pm \dfrac{1}{-|u| \sqrt{1- u^2}} \dfrac{dy}{dx} \\[3mm] 12. \dfrac{d}{dx}(\csc h^{-1}(u)) = \pm \dfrac{1}{-|u| \sqrt{1 + u^2}} \dfrac{du}{dx}$
Higher Order Derivatives
Let y = f(x) we have:

Second derivative is : $\dfrac{d}{dx} \left(\dfrac{du}{dx} \right) = \dfrac{d^2 u}{dx^2} = f^{||}(x) = u^{||}$
Third Derivative is: $\dfrac{d}{dx}\left(\dfrac{d^2u}{dx^2} \right) = \dfrac{d^3u}{dx^3} = f^{||}(x) = u^{||}$
nth Derivative is: $\dfrac{d}{dx} \left(\dfrac{d^{n- 1}u}{dx^{n- 1}} \right) = \dfrac{d^nu}{dx^n} = f^n(x) = u^n$
in some books, the following notation for higher derivatives is also used: $D^n(u) = \dfrac{d^nu}{dx^n}$
Higher Derivative formula for the product: Leibniz formula
Application of Derivatives
i. Recall

a. Volume of Cone(V) = $\dfrac{1}{3}\pi r^2h$ , r = radius, h= height

b. Volume of Cylinder (V) = $\pi r^2h$

c. Volume of Sphere(V) = $\dfrac{4}{3}\pi r^3$

d. Surface area of Sphere(S) = $4 \pi r^2$

e. Curved surface of Cylinder(S) = $2 \pi rh$

f. Total surface area of Cylinder (S) = $2 \pi r(r + h)$

ii. Equation of tangent at $(x_1, y_1)$ is $y- y_1 = f'(x_1) (x-x_1)$

iii. Equation of normal at $(x_1, y_1)$ is $y- y_1 = - \dfrac{1}{f'(x_1)}(x-x_1)$

iv. Let $m_1$ and $m_2$ be the slope of tangents to the two curves and $\theta$ be the angle between them, then, $\tan \theta = \dfrac{m_1 - m_2}{1 + m_1 m_2}$

v. Approximate increase in y is $dy = f^|(x)dy$

vi. Actual increase in y in $\delta y = f (x + \delta x)- f (x)$

vii. $error percent = \dfrac{error}{y} \times 100$

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