Question

Find the derivative of the following. please show the steps when you answer :1) f(x) = 8xe^x2) y= 5xe^x^43) f(x)= x^8+5/x4) f(t)= te^11-6t5) g(p) = pIn(2p+3)6) z= (te^6t + e^5t)^77) w= 2y+y^2 / 7+y

Answer 1

Since,

`(d)/(dx)x^n = nx^(n-1)`

`(d)/(dx)(f(x).g(x)) = f(x).(d)/(dx)(g(x)) + g(x).(d)/(dx)(f(x))`

`(d)/(dx)((f(x))/(g(x)))=(g(x).f'(x) - f(x) g'(x))/((g(x))^2)`

1)
`y = 8x e^x`

Differentiating with respect to x,

`(dy)/(dx)=8( x * e^x + e^x) = 8(xe^x + e^x) = 8e^x(x+1)`

2)
`y = 5x e^(x^4)`

Differentiating w. r. t x,

`(dy)/(dx)=5(x* 4x^3 e^(x^4)+e^(x^4))=5e^(x^4)(4x^4+1)`

3)
`y = x^8 + (5)/(x^4)`

Differentiating w. r. t. x,

`(dy)/(dx)=8x^7 - (5)/(x^5)* 4 = 8x^7 - (20)/(x^5)=(8x^(12)-20)/(x^5)`

4)
`f(t) = te^(11)-6t^5`

Differentiating w. r. t. t,

`f'(t) = e^(11) - 30t^4`

5)
`g(p) = p\ln(2p+3)`

Differentiating w. r. t. p,

`g'(p) = p(1)/(2p+3)(2) + \ln(2p+3) = (2p)/(2p+3)+\ln(2p+3)`

6)
`z = (te^(6t)+e^(5t))^7`

Differentiating w. r. t. t,

`(dz)/(dt)=7(te^(6t)+e^(5t))^6 ( 6te^(6t)+e^(6t) + 5e^(5t))`

7)
`w =(2y + y^2)/(7+y)`

Differentiating w. r. t. y,

`(dw)/(dy) = ((7+y)(2+2y)-(2y+y^2))/((7+y)^2) = (14 + 2y + 14y +2y^2 - 2y - y^2)/((7+y)^2)=(14+14y+y^2)/((7+y)^2)`