Calculus Examples
dydx=yx+y√xydydx=yx+y√xy
Step 1
Step 1.1
Assume √y2=y√y2=y.
dydx=yx+√y2√xydydx=yx+√y2√xy
Step 1.2
Combine √y2√y2 and √xy√xy into a single radical.
dydx=yx+√y2xydydx=yx+√y2xy
Step 1.3
Reduce the expression y2xyy2xy by cancelling the common factors.
Step 1.3.1
Factor yy out of y2y2.
dydx=yx+√y⋅yxydydx=yx+√y⋅yxy
Step 1.3.2
Factor yy out of xyxy.
dydx=yx+√y⋅yyxdydx=yx+√y⋅yyx
Step 1.3.3
Cancel the common factor.
dydx=yx+√y⋅yyx
Step 1.3.4
Rewrite the expression.
dydx=yx+√yx
dydx=yx+√yx
dydx=yx+√yx
Step 2
Let V=yx. Substitute V for yx.
dydx=V+√V
Step 3
Solve V=yx for y.
y=Vx
Step 4
Use the product rule to find the derivative of y=Vx with respect to x.
dydx=xdVdx+V
Step 5
Substitute xdVdx+V for dydx.
xdVdx+V=V+√V
Step 6
Step 6.1
Separate the variables.
Step 6.1.1
Solve for dVdx.
Step 6.1.1.1
Move all terms not containing dVdx to the right side of the equation.
Step 6.1.1.1.1
Subtract V from both sides of the equation.
xdVdx=V+√V-V
Step 6.1.1.1.2
Combine the opposite terms in V+√V-V.
Step 6.1.1.1.2.1
Subtract V from V.
xdVdx=0+√V
Step 6.1.1.1.2.2
Add 0 and √V.
xdVdx=√V
xdVdx=√V
xdVdx=√V
Step 6.1.1.2
Divide each term in xdVdx=√V by x and simplify.
Step 6.1.1.2.1
Divide each term in xdVdx=√V by x.
xdVdxx=√Vx
Step 6.1.1.2.2
Simplify the left side.
Step 6.1.1.2.2.1
Cancel the common factor of x.
Step 6.1.1.2.2.1.1
Cancel the common factor.
xdVdxx=√Vx
Step 6.1.1.2.2.1.2
Divide dVdx by 1.
dVdx=√Vx
dVdx=√Vx
dVdx=√Vx
dVdx=√Vx
dVdx=√Vx
Step 6.1.2
Multiply both sides by 1√V.
1√VdVdx=1√V⋅√Vx
Step 6.1.3
Cancel the common factor of √V.
Step 6.1.3.1
Cancel the common factor.
1√VdVdx=1√V⋅√Vx
Step 6.1.3.2
Rewrite the expression.
1√VdVdx=1x
1√VdVdx=1x
Step 6.1.4
Rewrite the equation.
1√VdV=1xdx
1√VdV=1xdx
Step 6.2
Integrate both sides.
Step 6.2.1
Set up an integral on each side.
∫1√VdV=∫1xdx
Step 6.2.2
Integrate the left side.
Step 6.2.2.1
Apply basic rules of exponents.
Step 6.2.2.1.1
Use n√ax=axn to rewrite √V as V12.
∫1V12dV=∫1xdx
Step 6.2.2.1.2
Move V12 out of the denominator by raising it to the -1 power.
∫(V12)-1dV=∫1xdx
Step 6.2.2.1.3
Multiply the exponents in (V12)-1.
Step 6.2.2.1.3.1
Apply the power rule and multiply exponents, (am)n=amn.
∫V12⋅-1dV=∫1xdx
Step 6.2.2.1.3.2
Combine 12 and -1.
∫V-12dV=∫1xdx
Step 6.2.2.1.3.3
Move the negative in front of the fraction.
∫V-12dV=∫1xdx
∫V-12dV=∫1xdx
∫V-12dV=∫1xdx
Step 6.2.2.2
By the Power Rule, the integral of V-12 with respect to V is 2V12.
2V12+C1=∫1xdx
2V12+C1=∫1xdx
Step 6.2.3
The integral of 1x with respect to x is ln(|x|).
2V12+C1=ln(|x|)+C2
Step 6.2.4
Group the constant of integration on the right side as C.
2V12=ln(|x|)+C
2V12=ln(|x|)+C
Step 6.3
Solve for V.
Step 6.3.1
Divide each term in 2V12=ln(|x|)+C by 2 and simplify.
Step 6.3.1.1
Divide each term in 2V12=ln(|x|)+C by 2.
2V122=ln(|x|)2+C2
Step 6.3.1.2
Simplify the left side.
Step 6.3.1.2.1
Cancel the common factor.
2V122=ln(|x|)2+C2
Step 6.3.1.2.2
Divide V12 by 1.
V12=ln(|x|)2+C2
V12=ln(|x|)2+C2
Step 6.3.1.3
Simplify the right side.
Step 6.3.1.3.1
Simplify each term.
Step 6.3.1.3.1.1
Rewrite ln(|x|)2 as 12ln(|x|).
V12=12ln(|x|)+C2
Step 6.3.1.3.1.2
Simplify 12ln(|x|) by moving 12 inside the logarithm.
V12=ln(|x|12)+C2
V12=ln(|x|12)+C2
V12=ln(|x|12)+C2
V12=ln(|x|12)+C2
Step 6.3.2
Raise each side of the equation to the power of 2 to eliminate the fractional exponent on the left side.
(V12)2=(ln(|x|12)+C2)2
Step 6.3.3
Simplify the left side.
Step 6.3.3.1
Simplify (V12)2.
Step 6.3.3.1.1
Multiply the exponents in (V12)2.
Step 6.3.3.1.1.1
Apply the power rule and multiply exponents, (am)n=amn.
V12⋅2=(ln(|x|12)+C2)2
Step 6.3.3.1.1.2
Cancel the common factor of 2.
Step 6.3.3.1.1.2.1
Cancel the common factor.
V12⋅2=(ln(|x|12)+C2)2
Step 6.3.3.1.1.2.2
Rewrite the expression.
V1=(ln(|x|12)+C2)2
V1=(ln(|x|12)+C2)2
V1=(ln(|x|12)+C2)2
Step 6.3.3.1.2
Simplify.
V=(ln(|x|12)+C2)2
V=(ln(|x|12)+C2)2
V=(ln(|x|12)+C2)2
V=(ln(|x|12)+C2)2
Step 6.4
Simplify the constant of integration.
V=(ln(|x|12)+C)2
V=(ln(|x|12)+C)2
Step 7
Substitute yx for V.
yx=(ln(|x|12)+C)2
Step 8
Step 8.1
Multiply both sides by x.
yxx=(ln(|x|12)+C)2x
Step 8.2
Simplify.
Step 8.2.1
Simplify the left side.
Step 8.2.1.1
Cancel the common factor of x.
Step 8.2.1.1.1
Cancel the common factor.
yxx=(ln(|x|12)+C)2x
Step 8.2.1.1.2
Rewrite the expression.
y=(ln(|x|12)+C)2x
y=(ln(|x|12)+C)2x
y=(ln(|x|12)+C)2x
Step 8.2.2
Simplify the right side.
Step 8.2.2.1
Reorder factors in (ln(|x|12)+C)2x.
y=x(ln(|x|12)+C)2
y=x(ln(|x|12)+C)2
y=x(ln(|x|12)+C)2
y=x(ln(|x|12)+C)2