Differentiate the following functions: f(x) = x * tan2(1/x) and y = cos(sqrt(1 + sin2(x)))
|Mathematics
Differentiate the following functions: f(x) = x * tan2(1/x) and y = cos(sqrt(1 + sin2(x)))
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5 steps
Answer
tan2(1/x)−x2tan(1/x)sec2(1/x)
Here are the derivatives for each function:
To differentiate f(x)=x⋅tan2(1/x), we use the product rule f′(x)=u′v+uv′.
Let u=x and v=tan2(1/x).
Step 1: Find u′.
u′=dxd(x)=1
Step 2: Find v′ using the chain rule.
v=(tan(x−1))2v′=2tan(x−1)⋅dxd(tan(x−1))v′=2tan(x−1)⋅sec2(x−1)⋅dxd(x−1)v′=2tan(1/x)⋅sec2(1/x)⋅(−1x−2)v′=−x22tan(1/x)sec2(1/x)
Step 3: Apply the product rule.
f′(x)=1⋅tan2(1/x)+x⋅(−x22tan(1/x)sec2(1/x))f′(x)=tan2(1/x)−x2tan(1/x)sec2(1/x)
The derivative is:
tan2(1/x)−x2tan(1/x)sec2(1/x)
To differentiate y=cos(1+sin2(x)), we use the chain rule multiple times.
Step 1: Differentiate the outermost function, cos(u).
dxdy=−sin(1+sin2(x))⋅dxd(1+sin2(x))
Step 2: Differentiate 1+sin2(x)=(1+sin2(x))1/2.
dxd((1+sin2(x))1/2)=21(1+sin2(x))−1/2⋅dxd(1+sin2(x))
Step 3: Differentiate 1+sin2(x).
dxd(1+sin2(x))=0+2sin(x)cos(x)
Step 4: Combine the results.
dxdy=−sin(1+sin2(x))⋅21+sin2(x)1⋅(2sin(x)cos(x))dxdy=−1+sin2(x)sin(x)cos(x)sin(1+sin2(x))
The derivative is:
−1+sin2(x)sin(x)cos(x)sin(1+sin2(x))
To differentiate y=ln(1−x2xsin(x)), we first simplify using logarithm properties.
Step 1: Apply logarithm properties ln(a/b)=ln(a)−ln(b) and ln(ab)=ln(a)+ln(b).
y=ln(xsin(x))−ln(1−x2)y=ln(x)+ln(sin(x))−21ln(1−x2)
Step 2: Differentiate each term.
dxd(ln(x))=x1dxd(ln(sin(x)))=sin(x)1⋅cos(x)=cot(x)dxd(−21ln(1−x2))=−21⋅1−x21⋅(−2x)=1−x2x
Step 3: Combine the derivatives.
dxdy=x1+cot(x)+1−x2x
The derivative is:
x1+cot(x)+1−x2x
To differentiate y=sec2(sin(x)), we use the chain rule multiple times.
Step 1: Rewrite the function.
y=(sec((sin(x))1/2))2
Step 2: Apply the power rule and chain rule.
dxdy=2sec(sin(x))⋅dxd(sec(sin(x)))
Step 3: Differentiate sec(sin(x)).
dxd(sec(sin(x)))=sec(sin(x))tan(sin(x))⋅dxd(sin(x))
Step 4: Differentiate sin(x)=(sin(x))1/2.
dxd((sin(x))1/2)=21(sin(x))−1/2⋅dxd(sin(x))dxd((sin(x))1/2)=2sin(x)1⋅cos(x)=2sin(x)cos(x)
Step 5: Combine all parts.
dxdy=2sec(sin(x))⋅sec(sin(x))tan(sin(x))⋅2sin(x)cos(x)dxdy=2sin(x)2sec2(sin(x))tan(sin(x))cos(x)dxdy=sin(x)sec2(sin(x))tan(sin(x))cos(x)
The derivative is:
sin(x)sec2(sin(x))tan(sin(x))cos(x)
To differentiate y=ln(cos(5x2+3)), we first simplify using logarithm properties.
Step 1: Apply logarithm property ln(ab)=bln(a).
y=ln((cos(5x2+3))1/2)y=21ln(cos(5x2+3))
Step 2: Differentiate using the chain rule.
dxdy=21⋅cos(5x2+3)1⋅dxd(cos(5x2+3))
Step 3: Differentiate cos(5x2+3).
dxd(cos(5x2+3))=−sin(5x2+3)⋅dxd(5x2+3)
Step 4: Differentiate 5x2+3.
dxd(5x2+3)=10x
Step 5: Combine the results.
dxdy=21⋅cos(5x2+3)1⋅(−sin(5x2+3))⋅(10x)dxdy=−2cos(5x2+3)10xsin(5x2+3)dxdy=−5xtan(5x2+3)
The derivative is:
−5xtan(5x2+3)
That's 2 down. 3 left today — send the next one.
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This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the derivatives for each function: 6. To differentiate f(x) = x · ^2(1/x), we use the product rule f'(x) = u'v + uv'. Let u = x and v = ^2(1/x). Step 1: Find u'. u' = (d)/(dx)(x) = 1 Step 2: Find v' using the chain rule. v = ((x^-1))^2 v' = 2(x^-1) · (d)/(dx)((x^-1)) v' = 2(x^-1) · ^2(x^-1) · (d)/(dx)(x^-1) v' = 2(1/x) · ^2(1/x) · (-1x^-2) v' = -(2)/(x^2)(1/x)^2(1/x) Step 3: Apply the product rule. f'(x) = 1 · ^2(1/x) + x · (-(2)/(x^2)(1/x)^2(1/x)) f'(x) = ^2(1/x) - (2)/(x)(1/x)^2(1/x) The derivative is: ^2(1/x) - (2)/(x)(1/x)^2(1/x) 7. To differentiate y = (sqrt(1 + ^2(x))), we use the chain rule multiple times. Step 1: Differentiate the outermost function, (u). (dy)/(dx) = -(sqrt(1 + ^2(x))) · (d)/(dx)(sqrt(1 + ^2(x))) Step 2: Differentiate sqrt(1 + ^2(x)) = (1 + ^2(x))^1/2. (d)/(dx)((1 + ^2(x))^1/2) = (1)/(2)(1 + ^2(x))^-1/2 · (d)/(dx)(1 + ^2(x)) Step 3: Differentiate 1 + ^2(x). (d)/(dx)(1 + ^2(x)) = 0 + 2(x)(x) Step 4: Combine the results. (dy)/(dx) = -(sqrt(1 + ^2(x))) · (1)/(2sqrt(1 + ^2(x))) · (2(x)(x)) (dy)/(dx) = -(x)(x)(sqrt(1 + ^2(x)))sqrt(1 + ^2(x)) The derivative is: -(x)(x)(sqrt(1 + ^2(x)))sqrt(1 + ^2(x)) 8. To differentiate y = ((x(x))/(sqrt(1-x^2))), we first simplify using logarithm properties. Step 1: Apply logarithm properties (a/b) = (a) - (b) and (ab) = (a) + (b). y = (x(x)) - (sqrt(1-x^2)) y = (x) + ((x)) - (1)/(2)(1-x^2) Step 2: Differentiate each term. (d)/(dx)((x)) = (1)/(x) (d)/(dx)(((x))) = (1)/((x)) · (x) = (x) (d)/(dx)(-(1)/(2)(1-x^2)) = -(1)/(2) · (1)/(1-x^2) · (-2x) = (x)/(1-x^2) Step 3: Combine the derivatives. (dy)/(dx) = (1)/(x) + (x) + (x)/(1-x^2) The derivative is: (1)/(x) + (x) + (x)/(1-x^2) 9. To differentiate y = ^2(sqrt((x))), we use the chain rule multiple times. Step 1: Rewrite the function. y = ((((x))^1/2))^2 Step 2: Apply the power rule and chain rule. (dy)/(dx) = 2(sqrt((x))) · (d)/(dx)((sqrt((x)))) Step 3: Differentiate (sqrt((x))). (d)/(dx)((sqrt((x)))) = (sqrt((x)))(sqrt((x))) · (d)/(dx)(sqrt((x))) Step 4: Differentiate sqrt((x)) = ((x))^1/2. (d)/(dx)(((x))^1/2) = (1)/(2)((x))^-1/2 · (d)/(dx)((x)) (d)/(dx)(((x))^1/2) = (1)/(2sqrt((x))) · (x) = ((x))/(2sqrt((x))) Step 5: Combine all parts. (dy)/(dx) = 2(sqrt((x))) · (sqrt((x)))(sqrt((x))) · ((x))/(2sqrt((x))) (dy)/(dx) = 2^2(sqrt((x)))(sqrt((x)))(x)2sqrt((x)) (dy)/(dx) = ^2(sqrt((x)))(sqrt((x)))(x)sqrt((x)) The derivative is: ^2(sqrt((x)))(sqrt((x)))(x)sqrt((x)) 10. To differentiate y = (sqrt((5x^2 + 3))), we first simplify using logarithm properties. Step 1: Apply logarithm property (a^b) = b(a). y = (((5x^2 + 3))^1/2) y = (1)/(2)((5x^2 + 3)) Step 2: Differentiate using the chain rule. (dy)/(dx) = (1)/(2) · (1)/((5x^2 + 3)) · (d)/(dx)((5x^2 + 3)) Step 3: Differentiate (5x^2 + 3). (d)/(dx)((5x^2 + 3)) = -(5x^2 + 3) · (d)/(dx)(5x^2 + 3) Step 4: Differentiate 5x^2 + 3. (d)/(dx)(5x^2 + 3) = 10x Step 5: Combine the results. (dy)/(dx) = (1)/(2) · (1)/((5x^2 + 3)) · (-(5x^2 + 3)) · (10x) (dy)/(dx) = -(10x(5x^2 + 3))/(2(5x^2 + 3)) (dy)/(dx) = -5x(5x^2 + 3) The derivative is: -5x(5x^2 + 3) That's 2 down. 3 left today — send the next one.
