
Find the derivative of , by using the first principle of derivatives.
Answer
471.9k+ views
Hint: We recall the first principle of derivative. We assume a small change in as and its corresponding change in as . We find the average rate of change as . We take limit to find the instantaneous rate of change as derivative of .
Complete step-by-step solution:
We are given the function in the question. Let us have . Let be a very small change in and the corresponding change in be . So we have;
We subtract both sides of the above equation to have;
We divide both sides of the above step to have;
We take limit both sides of the above step to have;
We use the trigonometric identity for in the above step to have;
We use law of product if limits in the right hand side of the above step to have;
We use the standard limit for in the right hand side of the above step to have;
We use the double angle formula for in the right hand side of the above step to have;
We know from first principle of derivative that . So we have
Note: We can use chain rule to directly find the derivative of . If composite function is defined as then the chain rule is given as . We can also use the first principle for derivative with a very small change as . The derivative of the function at particular points geometrically gives the slope of the tangent to the curve of the function. The first principle is also known as the delta method.
Complete step-by-step solution:
We are given the function
We subtract
We divide
We take limit
We use the trigonometric identity
We use law of product if limits in the right hand side of the above step to have;
We use the standard limit
We use the double angle formula
We know from first principle of derivative that
Note: We can use chain rule to directly find the derivative of
Latest Vedantu courses for you
Grade 11 Science PCM | CBSE | SCHOOL | English
CBSE (2025-26)
School Full course for CBSE students
₹41,848 per year
Recently Updated Pages
Express the following as a fraction and simplify a class 7 maths CBSE

The length and width of a rectangle are in ratio of class 7 maths CBSE

The ratio of the income to the expenditure of a family class 7 maths CBSE

How do you write 025 million in scientific notatio class 7 maths CBSE

How do you convert 295 meters per second to kilometers class 7 maths CBSE

Write the following in Roman numerals 25819 class 7 maths CBSE

Trending doubts
State and prove Bernoullis theorem class 11 physics CBSE

What are Quantum numbers Explain the quantum number class 11 chemistry CBSE

Write the differences between monocot plants and dicot class 11 biology CBSE

1 ton equals to A 100 kg B 1000 kg C 10 kg D 10000 class 11 physics CBSE

State the laws of reflection of light

In northern hemisphere 21st March is called as A Vernal class 11 social science CBSE
