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bookIntroductions to Partial Derivatives

Partial derivatives measure how functions change with respect to one variable alone. To do this in practice, you're treating one variable as a constant, deriving only one variable at a time.

What Are Partial Derivatives?

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

∂f∂xlim⁡h→0f(x+h,y)−f(x,y)h∂f∂ylim⁡h→0f(x,y+h)−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, ∂f∂x\frac{\partial f}{\partial x}:

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

Let's compute, ∂f∂y\frac{\partial f}{\partial y}:

∂f∂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

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bookIntroductions to Partial Derivatives

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Partial derivatives measure how functions change with respect to one variable alone. To do this in practice, you're treating one variable as a constant, deriving only one variable at a time.

What Are Partial Derivatives?

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

∂f∂xlim⁡h→0f(x+h,y)−f(x,y)h∂f∂ylim⁡h→0f(x,y+h)−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, ∂f∂x\frac{\partial f}{\partial x}:

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

Let's compute, ∂f∂y\frac{\partial f}{\partial y}:

∂f∂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

Все було зрозуміло?

Як ми можемо покращити це?

Дякуємо за ваш відгук!

Секція 3. Розділ 7
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