Question

微分问题

y=sinx-cosx→?
y=2x-3cosx→?
y=sin(2x+1)→?
y=sin^4x→?
y=tan^4x→?
y=cos(sinx)→?

y=e^3x-5→?
y=e^2xe^x^2→?
y=x^2e^x→?
y=log(x^2-2)→?
y=2xlogx→?
y=(logx)^3→?

Answer 1

y = sinx - cosx
y' = cosx + sinx
dy = (cosx + sinx) dx

y = 2x - 3cosx
y' = 2 + 3sinx
dy = (2 + 3sinx) dx

y = sin(2x + 1)
y' = cos(2x + 1) • 2 = 2cos(2x + 1)
dy = 2cos(2x + 1) dx

y = sin⁴x
y' = 4sin³x • cosx = 4cosxsin³x
dy = 4cosxsin³x dx

y = tan⁴x
y' = 4tan³x • sec²x = 4sec²xtan³x
dy = 4sec²xtan³x dx

y = cos(sinx)
y' = - sin(sinx) • cosx = - cosxsin(sinx)
dy = - cosxsin(sinx) dx

y = e^(3x - 5)
y' = e^(3x - 5) • 3 = 3e^(3x - 5)
dy = 3e^(3x - 5) dx

y = e^(2x)e^(x²)
y' = e^(x²) • e^(2x) • 2 + e^(2x) • e^(x²) • 2x
= 2(x + 1)e^(2x)e^(x²)
dy = 2(x + 1)e^(2x)e^(x²) dx

y = x²e^x
y' = x² • e^x + e^x • 2x
= x(x + 2)e^x
dy = x(x + 2)e^x dx

y = log(x² - 2) = ln(x² - 2)/ln(10)
y' = 1/(x² - 2) • 2x • 1/ln(10)
= 2xln(10)/(x² - 2)
dy = 2xln(10)/(x² - 2) dx

y = 2xlogx = 2xlnx/ln(10)
y' = 2/ln(10) • (x • 1/x + lnx • 1)
= 2(1 + lnx)/ln(10)
dy = 2(1 + lnx)/ln(10) dx

y = (logx)³ = 3logx = 3lnx/ln(10)
y' = 3/ln(10) • 1/x
= 3/(xln(10))
dy = 3/(xln(10)) dx