Derivative problems?

Question

1.if y = x^(x+2), what is dy/dx?

2. if y= (x+5)^x, what is dy/dx.

3. if y = 3.14^x, what is dy/dx.

thank a lot for sharing ur time for my questions. i really wanna kno the answers exactly.

Answer 1

These require logarithmic differentiation.

Take the ln of both sides.

1. ln(y) = (x+2)*ln(x)
Differentiate with respect to x:

(1/y)*dy/dx = (x+2)*1/x + ln(x)

Multiply by y on both sides:

dy/dx = y[(x+2)/x + ln(x)]

Replace y: dy/dx = x^(x+2)[(x+2)/x + ln(x)]

2. ln(y) = x*ln(x+5)

(1/y)dy/dx = x*1/(x+5) + ln(x+5)

dy/dx = y[x/(x+5) + ln(x+5)

dy/dx = (x+5)^x[x/(x+5) + ln(x+5)]

3. Use the formula d/dx(a^u) = a^u*ln(a)*du/dx

Then dy/dx = 3.14^x*ln(3.14)

Answer 2

1. dy/dx=((x+2)/x+(x+2)*ln(x))*(x^(x+2))
2. dy/dx=(x/(x+5)+ln(x+5)) *((x+5)^x)
3. dy/dx=ln(3.14)*((3.14)^x)

comment: y^z can be written in this form exp(ln(y)*z).

Answer 3

1) ( log(x) + (x+2)/x ) * (x^( x+2 ))

2) ( log(x+5) + x/(x+5) ) * (( x+5 )^x)

3) ( 3.14 ^ x ) * log( 3.14 )