# JEE Main & Advanced Mathematics Differentiation Theorems for Differentiation

## Theorems for Differentiation

Category : JEE Main & Advanced

Let $f(x),\,g(x)$and $u(x)$be differentiable functions

(1) If at all points of a certain interval, ${f}'(x)=0,$ then the function $f(x)$ has a constant value within this interval.

(2) Chain rule

(i) Case I : If $y$ is a function of $u$ and $u$ is a function of $x,$ then derivative of $y$ with respect to $x$ is $\frac{dy}{dx}=\frac{dy}{du}\,\frac{du}{dx}$ or $y=f(u)$ $\Rightarrow \,\frac{dy}{dx}=f'(u)\frac{du}{dx}$.

(ii) Case II : If $y$ and $x$ both are expressed in terms of $t,\,\,y$ and $x$ both are differentiable with respect to $t,$ then $\frac{dy}{dx}=\frac{dy/dt}{dx/dt}$.

(3) Sum and difference rule: Using linear property $\frac{d}{dx}(f(x)\pm g(x))=\frac{d}{dx}(f(x))\pm \frac{d}{dx}(g(x))$

(4) Product rule

(i) $\frac{d}{dx}(f(x).g(x))=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x)$

(ii) $c\in (a,\,b)$

(5) Scalar multiple rule : $\frac{d}{dx}(k\,f(x))=k\frac{d}{dx}f(x)$

(6) Quotient rule : $\frac{d}{dx}\left( \frac{f(x)}{g(x)} \right)\,=\frac{g(x)\frac{d}{dx}(f(x))-f(x)\frac{d}{dx}(g(x))}{{{(g(x))}^{2}}}$

Provided $g(x)\ne 0$.

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