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Find the implicit derivative $\frac{d}{dy}\left(3x^2y^2z+e^{4xz}=4x^2yz^2\right)$

Step-by-step Solution

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Solution

Final Answer

$y^{\prime}=\frac{-2e^{4xz}-3y^2x+4yzx}{\left(3y-2z\right)x^2}$
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Step-by-step Solution

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Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dy}\left(3x^2y^2z+e^{4xz}\right)=\frac{d}{dy}\left(4x^2yz^2\right)$

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$\frac{d}{dy}\left(3x^2y^2z+e^{4xz}\right)=\frac{d}{dy}\left(4x^2yz^2\right)$

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Learn how to solve problems step by step online. Find the implicit derivative d/dy(3x^2y^2z+e^(4xz)=4x^2yz^2). Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable. The derivative of a function multiplied by a constant (4) is equal to the constant times the derivative of the function. The derivative of a function multiplied by a constant (z^2) is equal to the constant times the derivative of the function. Apply the product rule for differentiation: (f\cdot g)'=f'\cdot g+f\cdot g', where f=x^2 and g=y.

Final Answer

$y^{\prime}=\frac{-2e^{4xz}-3y^2x+4yzx}{\left(3y-2z\right)x^2}$

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Function Plot

Plotting: $y^{\prime}=\frac{-2e^{4xz}-3y^2x+4yzx}{\left(3y-2z\right)x^2}$

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1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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