Question

find dx/dy : f(x) = 5.9x+√67x^x-8/2x^7+9​

Answer 1

Answer :

dy/dx = 5.9 + √67x^x(1+logx)-28x⁶

Solution :

• f(x) = 5.9x + √67x^x- 8/2x⁷ +9
• dy/dx = f'(x)

Note :

• d/dx (x^x) = x^x(1+logx)

Proof :

• Let x^x = t
• Apply log on both sides ,
• xlogx = logt
• Differentiate on both sides
• x(logx)'+logx(x)' = (logt)'
• x(1/x)+logx = 1/t dt/dx
• 1+logx = 1/t dt/dx
• dt/dx = t(1+logx)
• dt/dx = x^x(1+logx)
• d/dx(x^x) = x^x(1+logx)

Solution :

• f(x) = 5.9x + √67x^x- 8/2x⁷ +9
• f'(x) = d/dx(5.9x + √67x^x- 8/2x⁷ +9)
• dy/dx = 5.9dx/dx + √67 d(x^x)dx - 8/2 d(x⁷)dx + d(9)/dx
• dy/dx = 5.9 + √67 x^x(1+logx) -8/2 7x⁶ + 0
• dy/dx = 5.9 + √67x^x(1+logx)-28x⁶