+118667 y= x^4(9x-1)^5
\(y= x^4(9x-1)^5\\ u=x^4 \qquad v=(9x-1)^5\\ u'=4x^3 \qquad v'=5(9x-1)^4*9\\ \mbox{Now use product rule}\\ y'=uv'+vu'\)
you can finish it.
With the second one you can use the product rule but the quotient use will be easier. :)
a)-
Find the derivative of the following via implicit differentiation: d/dx(y) = d/dx(x^4 (-1+9 x)^5) The derivative of y is y'(x): y'(x) = d/dx(x^4 (-1+9 x)^5) Use the product rule, d/dx(u v) = v ( du)/( dx)+u ( dv)/( dx), where u = x^4 and v = (9 x-1)^5: y'(x) = (9 x-1)^5 d/dx(x^4)+x^4 d/dx((-1+9 x)^5) Use the power rule, d/dx(x^n) = n x^(n-1), where n = 4: d/dx(x^4) = 4 x^3: y'(x) = x^4 (d/dx((-1+9 x)^5))+4 x^3 (-1+9 x)^5 Using the chain rule, d/dx((9 x-1)^5) = ( du^5)/( du) ( du)/( dx), where u = 9 x-1 and ( d)/( du)(u^5) = 5 u^4: y'(x) = 4 x^3 (-1+9 x)^5+5 (9 x-1)^4 d/dx(-1+9 x) x^4 Differentiate the sum term by term and factor out constants: y'(x) = 4 x^3 (-1+9 x)^5+d/dx(-1)+9 d/dx(x) 5 x^4 (-1+9 x)^4 The derivative of -1 is zero: y'(x) = 4 x^3 (-1+9 x)^5+5 x^4 (-1+9 x)^4 (9 (d/dx(x))+0) Simplify the expression: y'(x) = 4 x^3 (-1+9 x)^5+45 x^4 (-1+9 x)^4 (d/dx(x)) The derivative of x is 1: y'(x) = 4 x^3 (-1+9 x)^5+1 45 x^4 (-1+9 x)^4 Simplify the expression: y'(x) = 45 x^4 (-1+9 x)^4+4 x^3 (-1+9 x)^5 Expand the left hand side: Answer: | | y'(x) = 45 x^4 (-1+9 x)^4+4 x^3 (-1+9 x)^5
b)-
Find the derivative of the following via implicit differentiation: d/dx(y) = d/dx(2/(7-x)^4) The derivative of y is y'(x): y'(x) = d/dx(2/(7-x)^4) Factor out constants: y'(x) = 2 d/dx(1/(7-x)^4) Using the chain rule, d/dx(1/(7-x)^4) = d/( du)1/u^4 ( du)/( dx), where u = 7-x and ( d)/( du)(1/u^4) = -4/u^5: y'(x) = 2 (-4 d/dx(7-x))/(7-x)^5 Simplify the expression: y'(x) = -(8 (d/dx(7-x)))/(7-x)^5 Differentiate the sum term by term and factor out constants: y'(x) = -(8 d/dx(7)-d/dx(x))/(7-x)^5 The derivative of 7 is zero: y'(x) = -(8 (-(d/dx(x))+0))/(7-x)^5 Simplify the expression: y'(x) = (8 (d/dx(x)))/(7-x)^5 The derivative of x is 1: y'(x) = (1 8)/(7-x)^5 Expand the left hand side: y'(x) = 8/(7-x)^5 Factor the numerator and denominator of the right hand side: y'(x) = -8/(-7+x)^5 Cancel common terms in the numerator and denominator: Answer: | | y'(x) = -8/(-7+x)^5
