Differentiate the following functions a 4t Squareroot t b 5x

Differentiate the following functions: (a) 4/t^- Squareroot t (b) 5^x - 3e^-x (c) ln(ln r + 1) (d) y^10/y^-1 Find an equation of the tangent line to the graph of the function y = (x - 1) ln x at x = 0.5.

Solution

3)

(a)derivative of tⁿ=nt^n-1

d_/dt((4_/t⁷) -t)

=d_/dt(4t^-7 -t^1/2)

=(4*(-7)t^-7-1) -((1_/2)t^(1/2)-1)

=(-28t^-8) -((1_/2)t^-1/2)

=(-28_/t⁸) -((1_/2)(1_/t^1/2))

=-(28_/t⁸) -(1_/(2t))

(b)

d_/dx(5^x +3e^-x)

=d_/dx(5^x) +d_/dx(3e^-x)

=(5^x ln(5))+(3e^-x*-1)

=(5^x ln(5))-(3e^-x)

(c)

d_/dr(ln(lnr +1))

=(1_{/(lnr +1)})*((1_/r) +0)

=(1_{/r(lnr +1)})

(d)

quotient rule:(u_/v)\'=(u\'v-uv\')_/v²

d_/dy[y¹⁰_/(y-1)]

=[(10y^10-1(y-1)) -(y¹⁰(1-0))]_/(y-1)²

=[(10y⁹(y-1)) -y¹⁰]_/(y-1)²

=[10y¹⁰-10y⁹ -y¹⁰]_/(y-1)²

=[9y¹⁰-10y⁹]_/(y-1)²

=y⁹(9y-10)_/(y-1)²