Derivatives Class 12 Mathematics Notes has been updated according to the latest syllabus of 2080. It means the notes of derivatives chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of this chapter because we are here to serve you.
Chapter – 12
Derivatives
The concept of derivatives is a fundamental topic in calculus. A derivative represents the rate of change of a function with respect to its variable. If $latex y = f(x)$ is a function of $latex x$, the derivative of $latex y$ with respect to $latex x$ is denoted by $latex \frac{dy}{dx}$ or $latex f'(x)$. It is defined as:
$latex f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) – f(x)}{\Delta x}$
In simpler terms, the derivative measures how a function’s output changes as its input changes. For instance, if $latex f(x)$ represents the position of an object at time $latex x$, then $latex f'(x)$ gives the object’s velocity at time $latex x$.
Derivative Rules
Key derivative rules covered in Class 12 include:
- Power Rule: For any real number $latex n$, $latex \frac{d}{dx}(x^n) = nx^{n-1}$Sum and
- Difference Rule: If $latex u(x)$ and $latex v(x)$ are two functions, then $latex \frac{d}{dx}[u(x) \pm v(x)] = u'(x) \pm v'(x)$
- Product Rule: If $latex u(x)$ and $latex v(x)$ are two functions, then $latex \frac{d}{dx}[u(x) \cdot v(x)] = u'(x) \cdot v(x) + u(x) \cdot v'(x)$
- Quotient Rule: If $latex u(x)$ and $latex v(x)$ are two functions, then $latex \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{u'(x) \cdot v(x) – u(x) \cdot v'(x)}{[v(x)]^2}$
- Chain Rule: For composite functions, if $latex y = f(g(x))$, then $latex \frac{dy}{dx} = f'(g(x)) \cdot g'(x)$
Derivatives Class 12 Mathematics Notes PDF
This PDF will provide the solutions of every question from the 1st exercise of class 12 derivatives chapter. Since this chapter has only one exercises, you don’t have to search for other exercises.
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Derivatives of Hyperbolic Functions
Hyperbolic functions, such as $latex \sinh(x)$, $latex \cosh(x)$, $latex \tanh(x)$, and their inverses, have specific derivative rules:
Higher Order Derivatives
Higher-order derivatives refer to the derivatives of derivatives. For example, the second derivative of a function $latex f(x)$ is denoted by $latex \frac{d^2y}{dx^2}$ or $latex f”(x)$, which measures the rate of change of the rate of change (acceleration, in physical terms).
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions of the form $latex y = [f(x)]^{g(x)}$ or products and quotients of functions, where taking the natural logarithm of both sides simplifies the differentiation process.
Application of Derivatives
The derivative is used in various applications such as finding the slope of a curve at a point, determining the rate of change of quantities, and solving problems related to maxima and minima of functions, which is crucial in optimization problems.
The first derivative test and the second derivative test are common methods used to identify local maxima and minima of a function.
These topics build on the foundational concepts of differentiation and are essential for understanding more advanced calculus concepts and solving real-world problems.
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