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bookIntroductions to Partial Derivatives

Note
Definition

A partial derivative measures how a multivariable function changes with respect to one variable while keeping all other variables constant. It captures the rate of change along a single dimension within a multivariable system.

What Are Partial Derivatives?

A partial derivative is written using the symbol βˆ‚\partial instead of dd for regular derivatives. If a function f(x,y)f(x,y) depends on both xx and yy, we compute:

βˆ‚fβˆ‚xlim⁑hβ†’0f(x+h,y)βˆ’f(x,y)hβˆ‚fβˆ‚ylim⁑hβ†’0f(x,y+h)βˆ’f(x,y)h\frac{\partial f}{\partial x} \lim_{h \rarr 0} \frac{f(x + h, y) - f(x,y)}{h} \\[6pt] \frac{\partial f}{\partial y} \lim_{h \rarr 0} \frac{f(x, y + h) - f(x,y)}{h}
Note
Note

When differentiating with respect to one variable, treat all other variables as constants.

Computing Partial Derivatives

Consider the function:

f(x,y)=x2y+3y2f(x,y) = x^2y + 3y^2

Let's find, βˆ‚fβˆ‚x\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial x$}}:

βˆ‚fβˆ‚x=2xy\frac{\partial f}{\partial x} = 2xy
  • Differentiate with respect to xx, treating yy as a constant.

Let's compute, βˆ‚fβˆ‚y\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial y$}}:

βˆ‚fβˆ‚y=x2+6y\frac{\partial f}{\partial y} = x^2 + 6y
  • Differentiate with respect to yy, treating xx as a constant.
question mark

Consider the function:

f(x,y)=4x3y+5y2f(x,y) = 4x^3y + 5y^2

Now, compute the partial derivative with respect to yy.

Select the correct answer

Everything was clear?

How can we improve it?

Thanks for your feedback!

SectionΒ 3. ChapterΒ 7

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bookIntroductions to Partial Derivatives

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Note
Definition

A partial derivative measures how a multivariable function changes with respect to one variable while keeping all other variables constant. It captures the rate of change along a single dimension within a multivariable system.

What Are Partial Derivatives?

A partial derivative is written using the symbol βˆ‚\partial instead of dd for regular derivatives. If a function f(x,y)f(x,y) depends on both xx and yy, we compute:

βˆ‚fβˆ‚xlim⁑hβ†’0f(x+h,y)βˆ’f(x,y)hβˆ‚fβˆ‚ylim⁑hβ†’0f(x,y+h)βˆ’f(x,y)h\frac{\partial f}{\partial x} \lim_{h \rarr 0} \frac{f(x + h, y) - f(x,y)}{h} \\[6pt] \frac{\partial f}{\partial y} \lim_{h \rarr 0} \frac{f(x, y + h) - f(x,y)}{h}
Note
Note

When differentiating with respect to one variable, treat all other variables as constants.

Computing Partial Derivatives

Consider the function:

f(x,y)=x2y+3y2f(x,y) = x^2y + 3y^2

Let's find, βˆ‚fβˆ‚x\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial x$}}:

βˆ‚fβˆ‚x=2xy\frac{\partial f}{\partial x} = 2xy
  • Differentiate with respect to xx, treating yy as a constant.

Let's compute, βˆ‚fβˆ‚y\frac{\raisebox{1pt}{$\partial f$}}{\raisebox{-1pt}{$\partial y$}}:

βˆ‚fβˆ‚y=x2+6y\frac{\partial f}{\partial y} = x^2 + 6y
  • Differentiate with respect to yy, treating xx as a constant.
question mark

Consider the function:

f(x,y)=4x3y+5y2f(x,y) = 4x^3y + 5y^2

Now, compute the partial derivative with respect to yy.

Select the correct answer

Everything was clear?

How can we improve it?

Thanks for your feedback!

SectionΒ 3. ChapterΒ 7
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