Answer to Question #212388 in Calculus for Kenneth

Question #212388

1. Find the derivative of the following functions with respect to x.

(a) f(x) = x^-1/2 minus xe^x + 10^x + e^x^2 / 2x minus cube root of 15

(b) F(x) = e^x sinx + sec x + ln x^2

2. If y^5 + 4xy^3 + x^3 +1 =0, find dy / dx and determine its values at the point ( x, y) = ( 0, -1).



1
Expert's answer
2021-07-05T16:50:52-0400

1.

(a) "f(x)=x^\\frac{-1}{ 2}-xe^x+10^x+\\frac{e^{x^2}}{2x}-\\sqrt[3]{15};"


"\\frac{df}{dx}=-\\frac{1}{2}x^\\frac{-3}{2}-(x+1)e^x+10^xln10+\\frac{4x^2e^{x^2}-2e^{x^2}}{4x^2}="


"=-\\frac{1}{2}x^\\frac{-3}{2}-(x+1)e^x+10^xln10+\\frac{(2x^2-1)e^{x^2}}{2x^2}";


(b) "F(x)=e^xsinx+secx+lnx^2;"


"\\frac{dF}{dx}=e^xsinx+e^xcosx+\\frac{tanx}{cosx}+\\frac{2}{x}="


"=(sinx+cosx)e^x+\\frac{tanx}{cosx}+\\frac{2}{x};"

2.

We use the implicit function differentiation rule:


"5y^4\\frac{dy}{dx}+4y^3+12xy^2\\frac{dy}{dx}+3x^2=0;"


"\\frac{dy}{dx}(5y^4+12xy^2)=-(4y^3+3x^2);"


"\\frac{dy}{dx}=-\\frac{4y^3+3x^2}{5y^4+12xy^2} ."


At the point (x,y)=(0,-1) we have:


"\\frac{y}{dx}=-\\frac{4\\cdot (-1)^3+0}{5\\cdot (-1)^4+0}=\\frac{4}{5} ."


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