Use implicit differentiation to find the equation of the tangent line to the cur

Question

Use implicit differentiation to find the equation of the tangent line to the curve xy^3+xy=10 at the point (5,1). Write the equation for the tangent line in the form y=mx+b:

Let f(x)=4+12x^2. Find f(x).

f(x)=

(b) Let f(x)=e^(4+12x^2). Find f(x).

f(x)=

Let f(x)=(7+7x). Find f(x).

f(x)=

Extra if can help

A mass attached to a vertical spring has position function given by s(t)=5sin(2t) where t is measured in seconds and s in inches.
Find the velocity at time t=1. (Include units)
Find the acceleration at time t=1. (Include units)

Explanation / Answer

1)xy3+xy=10

differentiate with respect to x on both sides

(1y3+x3y2(dy/dx))+(1y +x(dy/dx)) =0

at the point (5,1)

(13+5*3*12(dy/dx))+(1*1 +5(dy/dx)) =0

(1+15(dy/dx))+(1 +5(dy/dx)) =0

20dy/dx =-2

dy/dx =-1/10

slope of tangent m=-1/10

tangent line in the form y=mx+b

y=(-1/10)x+b passes through (5,1)

1=(-1/10)5 +b

1=(-1/2) +b

b=3/2

y=(-1/10)x+ (3/2) is the equation of tangent

2)

f(x)=(4+12x2).

f(x)=(4+12x2)1/2.

d/dx f(g(x)) =f '(g(x))* g'(x) ,d/dx xn=nxn-1

f (x)=(1/2)(4+12x2)(1/2)-1*(0 +12*2x)

f (x)=(12x)(4+12x2)(-1/2)

f (x)=(12x)/(4+12x2)

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