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 determined 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
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
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)
2)
3)
4)
Solution:
"(i) - (ii)"
diff. w.r.t.
put
Hence, the answer is 10.
Example 5: If
[JEE Main 2023]
1)
2)
3)
4)
Solution:
From (i) and (ii)
Hence, the answer is the option (1).
Differentiation is an important concept of Calculus. Differentiation of a function at a point represents the slope of the tangent to the graph of the function at that point. The differentiation of a constant is zero. With the help of differentiation, we can find the rate of change of one quantity for another. The concept of differentiation is the cornerstone on which the development of calculus rests.
The rate of change of a quantity
The derivative of a constant value is
The derivative of the exponential function is
The derivative of the logarithm function is
The derivative of
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