Differentiation is one of the important parts of Calculus, which applies in measuring the rate of change in the function at a certain point. Mathematically, it forms a powerful tool by which slopes of functions, the maximum and minimum of functions can be dertermined and problems on motion, growth, and decay can be solved and many more. These concepts of differentiation have been broadly applied in branches of mathematics, physics, engineering, economics, and biology.
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In this article, we will cover the concept of Differentiation of Polynomial Functions. This concept falls under the broader category of Calculus, which is a crucial Chapter in class 12 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last five years of the JEE Main exam (from 2013 to 2023), a total of five questions have been asked on this concept, including three in 2020, and two in 2023.
Let
Derivative of a Polynomial Function
The derivates for all basic functions are,
So the derivative of a constant function is zero. Also since a constant function is a horizontal line the slope of a constant function is 0.
2.
For
Since,
we can write
Next, divide both sides by h:
Thus,
Finally,
Derivative of Exponential and Logarithmic Functions
3.
4.
5.
6.
The derivatives of the basic trigonometric functions are
Derivative of the Trigonometric Functions (sin/cos/tan)
1.
Similarly,
2.
3.
Derivative of the Trigonometric Functions (cot/sec/csc)
4.
5.
6.
Example 1: If
1)
2)
3)
4)
Solution:
So
Hence, the answer is the option 1.
Example 2: Let
1)
2)
3)
4)
Solution:
Hence, the answer is the option (3).
Example 3: Let
[JEE Main 2020]
1)
2)
3)
4)
Solution:
Here,
Hence, the answer is the option 1.
Example 4: Let
then
[JEE Main 2023]
1) 10
2) -1
3) 2
4) 4
Solution:
"(i) - (ii)"
diff. w.r.t.
put
Hence, the answer is 10.
Example 5: If
[JEE Main 2023]
1) 14
2) -1
3) 2
4) 4
Solution:
From (i) and (ii)
Hence, the answer is the option (1).
The rate of change of a quantity y concerning another quantity x is called the derivative or differential coefficient of
The derivative of a constant value is 0.
The derivative of the exponential function is
The derivative of the logarithm function is
The derivative of
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