Notice: This page requires JavaScript to function properly.
Please enable JavaScript in your browser settings or update your browser.Impara Introductions to Partial Derivatives | Mathematical Analysis
Mathematics for Data Science

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

Tutto è chiaro?

Come possiamo migliorarlo?

Grazie per i tuoi commenti!

Sezione 3. Capitolo 7

Chieda ad AI

expand

Chieda ad AI

ChatGPT

Chieda pure quello che desidera o provi una delle domande suggerite per iniziare la nostra conversazione

Awesome!

Completion rate improved to 1.89

bookIntroductions to Partial Derivatives

Scorri per mostrare il menu

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

Tutto è chiaro?

Come possiamo migliorarlo?

Grazie per i tuoi commenti!

Sezione 3. Capitolo 7
some-alt