Difference between revisions of "Differentiation Rules"
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If <math>y(x) = u(x)+v(x)</math> then <math>\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}</math>. | If <math>y(x) = u(x)+v(x)</math> then <math>\frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx}</math>. | ||
− | + | [[Product Rule]]: | |
If <math>y(x) = u(x) \cdot v(x)</math> then <math>\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}</math>. | If <math>y(x) = u(x) \cdot v(x)</math> then <math>\frac{dy}{dx} = u(x)\frac{dv}{dx} + v(x)\frac{du}{dx}</math>. | ||
− | + | [[Quotient Rule]]: | |
If <math>y(x) = \frac{u(x)}{v(x)}</math> then <math>\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}</math>. | If <math>y(x) = \frac{u(x)}{v(x)}</math> then <math>\frac{dy}{dx} = \frac{v(x)\frac{du}{dx} - u(x)\frac{dv}{dx}}{(v(x))^2}</math>. | ||
− | + | [[Chain Rule]]: | |
If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}</math>. | If <math>y(x) = u(v(x))</math> then <math>\frac{dy}{dx} = \frac{du}{dv}\cdot \frac{dv}{dx}</math>. | ||
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==Derivatives of Trig Functions== | ==Derivatives of Trig Functions== | ||
+ | |||
+ | '''Derivative of Sine''' | ||
+ | If <math>y(x) = \sin x</math>, then <math>\frac{dy}{dx} = \cos x</math>. | ||
+ | |||
+ | '''Derivative of Cosine''' | ||
+ | If <math>y(x) = \cos x</math>, then <math>\frac{dy}{dx} = -\sin x</math>. | ||
+ | |||
+ | '''Derivative of Tangent''' | ||
+ | If <math>y(x) = \tan x</math>, then <math>\frac{dy}{dx} = \sec^2 x</math>. Note that this follows from the Quotient Rule. | ||
+ | |||
+ | '''Derivative of Cosec''' | ||
+ | If <math>y(x) = \csc x</math>, then <math>\frac{dy}{dx} = -\csc(x)\cot(x)</math>. |
Latest revision as of 10:50, 4 June 2024
Differentiation rules are rules (actually, theorems) used to compute the derivative of a function in calculus. In what follows, all functions are assumed to be differentiable.
Basic Rules
Derivative of a Constant:
If is a constant function then
.
Sum Rule:
If then
.
Product Rule:
If then
.
Quotient Rule:
If then
.
Chain Rule:
If then
.
Power Rule:
If then
. For integer
this is just a consequence of the product and quotient rules and induction, but it can also be proven for all real numbers
, e.g. by using the extended Binomial Theorem.
Derivatives of Trig Functions
Derivative of Sine
If , then
.
Derivative of Cosine
If , then
.
Derivative of Tangent
If , then
. Note that this follows from the Quotient Rule.
Derivative of Cosec
If , then
.