Practice Problems
Find the first order partial derivatives for each.
\(\textbf{1)}\) \(f(x,y)=x^2+y^2+3\)
\(\textbf{2)}\) \(f(x,y)=3x^2y\)
\(\textbf{3)}\) \(f(x,y,z)=x^2z+4y^3z+3z^2\)
\(\textbf{4)}\) \(f(x,y)=x \sin y+\cos x\)
\(\textbf{5)}\) \(f(x,y)=x^{5} \ln y -x\)
\(\textbf{6)}\) \(f(x,y)=e^{5x^2+2y}\)
\(\textbf{7)}\) \(f(x,y)=4x^4\sqrt[3]{y^2}-xy\)
See Related Pages
\(\bullet\text{ Partial Derivative Calculator (Symbolab)}\)
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\(\bullet\text{ Calculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Definition of Derivative}\)
\(\,\,\,\,\,\,\,\, \displaystyle \lim_{\Delta x\to 0} \frac{f(x+ \Delta x)-f(x)}{\Delta x} \)
\(\bullet\text{ Implicit Differentiation}\)
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\(\bullet\text{ Horizontal Tangent Line}\)
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\(\bullet\text{ Mean Value Theorem}\)
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\(\bullet\text{ Related Rates}\)
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\(\bullet\text{ Increasing and Decreasing Intervals}\)
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\(\bullet\text{ Intervals of concave up and down}\)
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\(\bullet\text{ Inflection Points}\)
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\(\bullet\text{ Graph of f(x), f'(x) and f”(x)}\)
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\(\bullet\text{ Newton’s Method}\)
\(\,\,\,\,\,\,\,\,x_{n+1}=x_n – \displaystyle \frac{f(x_n)}{f'(x_n)}\)
In Summary
First-order partial derivatives are a fundamental concept in calculus that involves taking the derivative of a function with respect to one of its variables, while keeping the other variables constant. This allows us to understand how a function changes as we vary one of its inputs.
Start with a function with multiple independent variables. For example z = f(x, y) has two independent variables x and y. If we keep y as a constant and take the derivative of z with respect to x, it is called a partial derivative and is denoted \(\frac{\partial z}{\partial x}\) or \(f_x(x,y)\).