Question 366556: Evening All,
Right its been a while since i have attempted maths, and have just gone back to further ed. I have attempted the following questions, which im sure should be simple, but of course like anything its just getting back into things after a number of years. i have listed the questions below, and my answers if anybody will take the time to have a quick look and correct me where neccessary i would appreciate this.
1)Find dy/dx by differentiating with respect to x the following expressions.
a) y=x^3-6x^2+9x-1 = 3x^2-12x+8
b) y=1/2x -sqrtx = 1/2-1/2x^-1/2
c) y=e^x-e^-x = e^x+e^-x
d) y=25cosx--sinx = -25sinx-cosx
e) y= 3sinhx-4coshx = 3hsinhx+4hsinhx
f) y= (x^2+1) sinhx = 1
2 Use product rule to obtain dy/dx for the following
a) y=x^2e^x = 2x(e^x)+x^2(e^x)
b) y= xtanx = 1(tanx)+x(sec^2x)
c) y= x^4ln(x)= 4x^3(ln(x))+x^4(1/x)
d) y= x^2sinx = 2x(sinx)+x^2(cosx)
e) y= e^-xcosx = -e^-x(cos x)+ e^-x (-sinx)
f) y=(x^2+1)sinhx = (2x +1) (sinhx)+(x^2+1)(cosh x)
Finally, use the quotient rule to differentiate with respect to x, simplifying as far as possible.
a) y= sinx/1+e^-x = 1+e^-x(cosx)-(1-e^x)sinx/(1+e^-x)^2
b) y= lnx/1+x^2 = 1+x^2(1/x)-1+2x(ln x)/(1+x^2)^2
Really this would be appreciated thank you
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! 1)Find dy/dx by differentiating with respect to x the following expressions.
a) y=x^3-6x^2+9x-1 = 3x^2-12x+8 *** + 9
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b) y=1/2x -sqrtx = 1/2-1/2x^-1/2 = (x - sqrt(x))/2x (same answer)
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c) y=e^x-e^-x = e^x+e^-x
d) y=25cosx--sinx = -25sinx-cosx ** there are 2 minus signs??
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e) y= 3sinhx-4coshx = 3hsinhx+4hsinhx Is it sin(hx), or hyperbolic?
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f) y= (x^2+1) sinhx = 1
2 Use product rule to obtain dy/dx for the following
a) y=x^2e^x = 2x(e^x)+x^2(e^x)
b) y= xtanx = 1(tanx)+x(sec^2x)
c) y= x^4ln(x)= 4x^3(ln(x))+x^4(1/x) = 4x^3*ln(x) + x^3
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d) y= x^2sinx = 2x(sinx)+x^2(cosx)
e) y= e^-xcosx = -e^-x(cos x)+ e^-x (-sinx)
f) y=(x^2+1)sinhx = (2x +1) (sinhx)+(x^2+1)(cosh x)
Not (2x + 1) *****
= 2x*sinh(x) + (x^2+1)*cosh(x)
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Finally, use the quotient rule to differentiate with respect to x, simplifying as far as possible.
a) y= sinx/1+e^-x = 1+e^-x(cosx)-(1-e^x)sinx/(1+e^-x)^2
b) y= lnx/1+x^2 = 1+x^2(1/x)-1+2x(ln x)/(1+x^2)^2
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