Neetesh Kumar | December 3, 2024
Calculus Homework Help
This is the solution to Math 1D
Assignment: 14.3 Question Number 16
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Step-by-step solution:
We are given the function:
f(x,y)=∫yxcos(et)dt
This is a definite integral with variable limits. To find the partial derivatives with respect to x and y, we apply the Leibniz rule for differentiation of integrals with variable limits.
Step 1: Find fx(x,y)
To find fx(x,y), we differentiate f(x,y) with respect to the upper limit x. By the Leibniz rule:
∂x∂∫yxcos(et)dt=cos(ex).
Thus:
fx(x,y)=cos(ex).
Step 2: Find fy(x,y)
To find fy(x,y), we differentiate f(x,y) with respect to the lower limit y. By the Leibniz rule, when differentiating with respect to the lower limit, we introduce a negative sign:
∂y∂∫yxcos(et)dt=−cos(ey).
Thus:
fy(x,y)=−cos(ey).
Final Answer:
fx(x,y)=cos(ex)
fy(x,y)=−cos(ey)
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