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Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)?
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Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – ...
Chain Rule Application:
The function g(u, v) can be thought of as a composition of functions, where the outer function is f and the inner function is u^2 - v^2. To find the partial derivatives of g with respect to u and v, we need to apply the chain rule.
Partial Derivative with respect to u:
To find ∂g/∂u, we differentiate g with respect to u treating v as a constant. Using the chain rule, we get:
∂g/∂u = ∂f/∂x * ∂(u^2 - v^2)/∂u
= f'(u^2 - v^2) * 2u
Partial Derivative with respect to v:
Similarly, to find ∂g/∂v, we differentiate g with respect to v treating u as a constant. Using the chain rule, we get:
∂g/∂v = ∂f/∂x * ∂(u^2 - v^2)/∂v
= f'(u^2 - v^2) * (-2v)
Twice Differentiable Function:
Since f is given to be twice differentiable, this implies that f'(u^2 - v^2) is also differentiable, which allows us to find the second-order partial derivatives of g.
Therefore, the correct answer is a) f'(u^2 - v^2) * 2u and f'(u^2 - v^2) * (-2v) for ∂g/∂u and ∂g/∂v respectively.
The function g(u, v) can be thought of as a composition of functions, where the outer function is f and the inner function is u^2 - v^2. To find the partial derivatives of g with respect to u and v, we need to apply the chain rule.
Partial Derivative with respect to u:
To find ∂g/∂u, we differentiate g with respect to u treating v as a constant. Using the chain rule, we get:
∂g/∂u = ∂f/∂x * ∂(u^2 - v^2)/∂u
= f'(u^2 - v^2) * 2u
Partial Derivative with respect to v:
Similarly, to find ∂g/∂v, we differentiate g with respect to v treating u as a constant. Using the chain rule, we get:
∂g/∂v = ∂f/∂x * ∂(u^2 - v^2)/∂v
= f'(u^2 - v^2) * (-2v)
Twice Differentiable Function:
Since f is given to be twice differentiable, this implies that f'(u^2 - v^2) is also differentiable, which allows us to find the second-order partial derivatives of g.
Therefore, the correct answer is a) f'(u^2 - v^2) * 2u and f'(u^2 - v^2) * (-2v) for ∂g/∂u and ∂g/∂v respectively.
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Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)?
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Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)?.
Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f : R → R be a twice differentiable function. If g(u, v) = f(u2 – v2), then a) b) c) d)?.
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