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Derivative of sum of two functions is sum of the derivatives of the functions. If , f and g be two functions such that their derivatives are defined in ______.
- a)Common domain
- b)Their individual domains
- c)Universally
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
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Derivative of sum of two functions is sum of the derivatives of the fu...
- The process of determining the derivative of a function is known as differentiation. It is clearly visible that the basic concept of derivative of a function is closely intertwined with limits. Therefore, it can be expected that the rules of derivatives are similar to that of limits.
- The following rules are a part of algebra of derivatives:
Consider f and g to be two real-valued functions such that the differentiation of these functions is defined in a common domain.
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Derivative of sum of two functions is sum of the derivatives of the fu...
**Explanation:**
The derivative of a function represents the rate at which the function is changing at a given point. It measures the rate of change of the function with respect to its independent variable.
According to the given statement, the derivative of the sum of two functions is equal to the sum of the derivatives of the individual functions. Mathematically, this can be expressed as:
(d/dx) [f(x) + g(x)] = f'(x) + g'(x)
Where f'(x) and g'(x) represent the derivatives of functions f(x) and g(x) respectively.
To understand why the correct answer is option 'A', let's consider an example:
Suppose we have two functions:
f(x) = x^2
g(x) = sin(x)
The domain of f(x) is all real numbers, as x can take any real value.
The domain of g(x) is also all real numbers, as sin(x) is defined for all real values of x.
Now, let's find the derivatives of f(x) and g(x) using the power rule and chain rule respectively:
f'(x) = 2x
g'(x) = cos(x)
Now, let's find the derivative of the sum of f(x) and g(x):
(d/dx) [f(x) + g(x)] = (d/dx) [x^2 + sin(x)]
= 2x + cos(x)
As we can see, the derivative of the sum of f(x) and g(x) is equal to the sum of their derivatives, which is consistent with the given statement.
Therefore, the correct answer is option 'A' - the derivatives of f(x) and g(x) are defined in the common domain, which in this case is all real numbers.
The derivative of a function represents the rate at which the function is changing at a given point. It measures the rate of change of the function with respect to its independent variable.
According to the given statement, the derivative of the sum of two functions is equal to the sum of the derivatives of the individual functions. Mathematically, this can be expressed as:
(d/dx) [f(x) + g(x)] = f'(x) + g'(x)
Where f'(x) and g'(x) represent the derivatives of functions f(x) and g(x) respectively.
To understand why the correct answer is option 'A', let's consider an example:
Suppose we have two functions:
f(x) = x^2
g(x) = sin(x)
The domain of f(x) is all real numbers, as x can take any real value.
The domain of g(x) is also all real numbers, as sin(x) is defined for all real values of x.
Now, let's find the derivatives of f(x) and g(x) using the power rule and chain rule respectively:
f'(x) = 2x
g'(x) = cos(x)
Now, let's find the derivative of the sum of f(x) and g(x):
(d/dx) [f(x) + g(x)] = (d/dx) [x^2 + sin(x)]
= 2x + cos(x)
As we can see, the derivative of the sum of f(x) and g(x) is equal to the sum of their derivatives, which is consistent with the given statement.
Therefore, the correct answer is option 'A' - the derivatives of f(x) and g(x) are defined in the common domain, which in this case is all real numbers.
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Derivative of sum of two functions is sum of the derivatives of the functions. If , f and g be two functions such that their derivatives are defined in ______.a)Common domainb)Their individual domainsc)Universallyd)None of the aboveCorrect answer is option 'A'. Can you explain this answer?
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