Find Derivative d(y)/dx of xmyn = (x + y)(m+n)
In this problem, we find the derivative of xmyn = (x + y)(m+n)
with respect to x. Since both x and y are involved, we use logarithmic and implicit differentiation.
Given Equation
xmyn = (x + y)(m + n)
Step 1: Take Logarithm on Both Sides
Taking log on both sides:
log(xmyn) = log[(x + y)(m+n)]
Using log laws:
m log x + n log y = (m + n) log(x + y)
Step 2: Differentiate with Respect to x
Differentiating both sides:
m(1/x) + n(1/y)(dy/dx) = (m + n)(1/(x + y))(1 + dy/dx)
Step 3: Solve for dy/dx
Rearranging terms and solving for dy/dx, we get:
dy/dx =
[ (m + n)y − n(x + y) ] / [ (m + n)x − m(x + y) ]
Logarithmic differentiation is especially useful when variables appear as powers and products.
Find Derivative d(y)/dx of xmyn = (x + y)(m+n)
Keywords
derivative of x^m y^n = (x + y)^(m+n), logarithmic differentiation, implicit differentiation, dy/dx, calculus, Class 11 math, Class 12 math, differentiation problems, exam calculus, math notes
Tags
calculus, derivative, logarithmic differentiation, implicit differentiation, Class 11, Class 12, math notes, exam question, differentiation, aklearningnepal, education,Find Derivative d(y)/dx of xmyn = (x + y)(m+n)
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Find Derivative d(y)/dx of xmyn = (x + y)(m+n)