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Rules of Differentiation (Chain Rule) is considered one of the most asked concept.
46 Questions around this concept.
Suppose $f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}$. Then the value of $f^{\prime}(0)$ is equal to
Let $y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1<x<1$. Then at $x=\frac{1}{2}$, the value of $225\left(y^{\prime}-y^{\prime \prime}\right)$ is equal to
If then
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Let $\mathrm{f}: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y, f(y) \neq 0$. If $f^{\prime}(1)=2024$, then
Rules of Differentiation (Sum/Difference/Product)
Let f(x) and g(x) be differentiable functions and k be a constant. Then each of the following rules hold:
Sum Rule
The derivative of the sum of a function f and a function g is the same as the sum of the derivative of f and the derivative of g.
In general,
Difference Rule
The derivative of the difference of a function f and a function g is the same as the difference of the derivative of f and the derivative of g.
Constant Multiple Rule
The derivative of a constant k multiplied by a function f is the same as the constant multiplied by the derivative of f
Product rule
Let f(x) and g(x) be differentiable functions. Then,
This means that the derivative of a product of two functions is the derivative of the first function times the second function plus the derivative of the second function times the first function
Extending the Product Rule
If 3 functions are involved, i.e let k(x) = f(x)・ g(x)・h(x)
Let us have a function k(x) as the product of the function f(x)g(x) and the function h(x). That is, k(x) = (f(x)・ g(x))・h(x). Thus,
Rules of Differentiation (Division or Quotient Rule)
Quotient Rule
Let f(x) and g(x) be differentiable functions. Then
OR
As we see in the following theorem, the derivative of the quotient is not the quotient of the derivatives.
If and
are differentiable funcitons, then
or
is also differentiable.
If , then
is known as chain rule. Or,
If and
then
The chain rule can be extended as follows:
If then
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