Find the derivatives of the following functions

y equals x Superscript StartFraction 11 Over 5 EndFraction

y'=

f left-parenthesis x right-parenthesis equals StartFraction 1 Over x Superscript 16 EndFraction

f'(x)=

y equals x Superscript 2 Baseline plus 9 x plus 3

y'=

f left-parenthesis z right-parenthesis equals negative StartFraction 1 Over z Superscript 3.9 EndFraction

f'(z)=

y equals RootIndex 14 StartRoot x EndRoot

dy/dx=

f left-parenthesis x right-parenthesis equals StartRoot StartFraction 1 Over x cubed EndFraction EndRoot

f'(x)=

y equals 7t squared plus 10e Superscript t

y'=

g left-parenthesis x right-parenthesis equals left-parenthesis 2 x Superscript 3 Baseline plus 1 right-parenthesis Superscript 5

g'(x)=

w equals left-parenthesis t Superscript 5 Baseline plus 1 right-parenthesis Superscript 92

w'=

y equals StartRoot s Superscript 11 Baseline plus 2 EndRoot

y'=

y equals 2 e Superscript 4 t plus 3

dy/dt=

y equals l n left-parenthesis 2 t plus 3 right-parenthesis

dy/dt

f left-parenthesis x right-parenthesis equals ? l n ? left-parenthesis e Superscript x Baseline plus 5 right-parenthesis

f'(x)=

please answer the whole question not parts of it. Thank you!

The question is:

Find the derivatives of the following functions

(1) y = x^(11/5)

(2) f(x) = 1/(x^16)

(3) y = x²(9x + 3)

(4) f(z) = -1/(z^3.9)

(5) y = (√x)^(1/14)

(6) f(x) = √(1/x³)

(7) y = 7t² + 10e^t

(8) g(x) = (2x³ + 1)^5

(9) w = (t^5 + 1)^92

(10) y = √(s^11 + 2)

(11) y = 2e^(4t + 3)

(12) y = ln(2t + 3)

(13) f(x) = ln(e^x + 5)

Step-by-step explanation:

If y = x^n, the derivative of y with respect to x is written as dy/dx or y', and it is given as

y' = nx^(n - 1).

If y = f(u), and u = u(x),

y' = f'(u) × u'(x)

If y = e^x, y' = e^x

If y = ln(u) and u = u(x)

y' = (1/u) × du/dx

Now, let us solve the problems given.

(1) y = x^(11/5)

y' = (11/5)x^(11/5 - 1)

= (11/5)x^(6/5)

(2) f(x) = 1/(x^16)

This can be written as

f(x) = x^(-16)

f'(x) = -16x^(-16 - 1)

= -16x^(-17)

(3) y = x²(9x + 3)

y' = 2x(9x + 3) + x²(9)

= 18x² + 6x + 9x²

= 27x² + 6x

(4) f(z) = -1/(z^3.9)

This can be written as

f(z) = z^(-3.9)

f'(z) = -3.9z^(-3.9 - 1)

= -3.9z^(-4.9)

(5) y = (√x)^(1/14)

This can be written as

y = x^(1/28)

y' = (1/28)x^(1/28 - 1)

= (1/28) x^(-27/28)

(6) f(x) = √(1/x³)

This can be written as

f(x) = x^(-3/2)

f'(x) = (-3/2)x^(-3/2 - 1)

= (-3/2)x^(-5/2)

(7) y = 7t² + 10e^t

y' = 14t + 10e^t

(8) g(x) = (2x³ + 1)^5

g'(x) = 6x² × 5(2x³ + 1)^4

= 30x²(2x³ + 1)^4

(9) w = (t^5 + 1)^92

w' = 5t × 92(t^5 + 1)^91

= 460t(t^5 + 1)^91

(10) y = √(s^11 + 2)

This can be written as

y = (s^11 + 2)^(1/2)

y = 11s × (1/2)(s^11 + 2)^(-1/2)

= (11s/2)(s^11 + 2)^(-1/2)

(11) y = 2e^(4t + 3)

y' = 4 × 2e^(4t + 3)

= 8e^(4t + 3)

(12) y = ln(2t + 3)

y' = 2/(2t + 3)

(13) f(x) = ln(e^x + 5)

f'(x) = e^(x)/(e^x + 5)

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