Trig functions
Given: f(x) = 3tan(x), determine f'(x).
f'(x) = 3sec2(x)
Determine the derivative of y = sqrt(x)-cuberoot(x2).
Write your answer with positive exponents.
dy/dx =1/(2 sqrt(x))- 2/(3cuberoot(x))
Determine f'(x) for: f(x) = 2pi
0
Given f(x) = 2/(2x4-5), determine f'(x).
f'(x) = -16x3/(2x4-5)2
Given f(x) = -x3(3x4 - 2), find the second derivative, f''(x), of f(x).
f''(x) = -126x5 + 12x
What is the derivative of y = sec(x)?
dy/dx = sec(x)tan(x)
Given f(x) = 7x3 - 3/x2 + 12, determine f'(1) and use this to determine if f(x) is increasing, decreasing or neither at x = 1.
f'(1) = 18
f(x) is increasing at x = 1 since f'(1) > 0.
Given f(x) = 4x4 − 5x − 3e2, determine f '(x),
f '(x) = 16x3− 5
Given y = (lnx)/(3x3-2x), determine dy/dx.
dy/dx = ((3x3-2x)(1/x)-(lnx)(9x2-2))/(3x3-2x)2
or simplified
dy/dx = ((3x2-2)-(lnx)(9x2-2))/(3x3-2x)2
Determine f'(x): f(x) = (5x5+5)(-2x5-3)
and simplify your answer.
f'(x) = -100x9 - 125x4
Given f(x)= x2 - sin(x), then determine f'(x) and f'(pi).
f'(x) = 2x - cos(x)
f'(pi) = 2pi + 1
Determine the slope of a line normal to the tangent line at x = -1 for f(x) = -(x + 7x2)
-1/13
Given f(x)=3/x3 , then determine f'(x). Use it to write the equation of the tangent line at x = -1.
f'(x)= -9/x4
The equation of the tangent line at x = -1 is
y +3 = -9(x + 1)
Given y = 2cosx/ex determine dy/dx and then name the slope of the tangent line at x = 0.
dy/dx = (-2exsinx - 2excosx)/e2x
The slope of the tangent line at x = 0 is -2.
Write the function for the instantaneous rate of change of f(x) = 2x3sinx.
IROC = 2x3cosx+ 6x2sinx
If f(x)=sin(x)cos(x), then determine f'(x) and f'(pi/3).
f'(x)=cos2(x)-sin2(x)
or f'(x)=-sin2(x)+cos2(x)
f'(pi/3) = -1/2
Determine the equation of a line tangent line to f(x) at x = 2 and f(x) = (3x - 7)2.
y - 1 = -6(x - 2)
Given y = (4x3 - 2x2 + 10)/(2x), determine dy/dx. Then determine the slope of the tangent line at x = -1.
dy/dx = 4x -1 -5x-2
The slope of the tangent line at x = -1 is -10.
Given g(x) = 2x3/lnx, determine g'(x).
g'(x) = (6x2lnx - 2x2)/(lnx)2
Determine the slope of the tangent when x = pi
y = cos(x)(1 + x2)
y' = -sin(x)(1+x2)+cos(x)(2x)
slope of the tangent when x = pi is -2pi
If f(x)=cot(x)/(sin(x)+1), then determine f'(x).
f'(x)=((-csc2(x)(sin(x)+1))-(cos(x)cot(x))) /(sin(x)+1)2
Determine the value(s) of x at which the function
f(x) = 4x + 8cosx has a horizontal tangent on the interval [0, 2pi).
At x = pi/6 and x = 5pi/6, f(x) has horizontal tangents.
Given f(x) = (1/3)x3 - 3x2 + 8x + 25, determine at which x-value(s) where f(x) has a horizontal tangent.
Set f'(x) = 0
At x = 2 and x = 4, f(x) has horizontal tangents.
Given h(x) = csc x/(4x2), determine h'(x) and simplify your answer.
h'(x) = (-4x2(csc x)(cot x) - 8xcsc x)/(16x4)
simplified
h'(x) = (-x(csc x)(cot x) - 2csc x)/(4x3)
Write the equation of the tangent line to
f(x) = (0.5x2 - 10)(lnx) at x = 1.
Then use equation of the tangent line to estimate f(1.01).
Equation of the tangent line
y = -9.5(x - 1)
f(1.01) is approximately -0.095