# Partial Derivatives: Solving Difficult Problems

• MHB
• Kamo123
In summary: So for the first one, you have\begin{align*}\pd{m}{q}&=\pd{}{q}\left[\ln(qh-2h^2)+2e^{(q-h^2+3)^4}-7\right] \\&=\frac{1}{qh-2h^2} \, \pd{}{q}(qh-2h^2)+2e^{(q-h^2+3)^4}\,\pd{}{q}\left[(q-h^2+3)^4\right].\end{align*}Can you continue?
Kamo123
Hello

I'm currently trying to solve these two problems:

1) Find the partial derivatives ∂m/∂q and ∂m/∂h of the function:

m=ln(qh-2h^2)+2e^(q-h^2+3)^4-7

Here, I know I should differentiate m with respect to q while treating h as a constant and vice versa. But I'm still stuck, and I'm not sure how to actually do it.

2) Find the partial derivative ∂z/∂x of the function:

(z+1)^3y-(zx)^2-3=x/z-ln(xyz)-5xy^2

I have tried some implicit differentiation here, but I can't really make it work out.

Please explain how to solve the two problems very explicitly.

Lillery said:
Hello

I'm currently trying to solve these two problems:

1) Find the partial derivatives ∂m/∂q and ∂m/∂h of the function:

m=ln(qh-2h^2)+2e^(q-h^2+3)^4-7

Here, I know I should differentiate m with respect to q while treating h as a constant and vice versa. But I'm still stuck, and I'm not sure how to actually do it.

So you have
$$m=\ln(qh-2h^2)+2e^{(q-h^2+3)^4}-7.$$
Is this correct?

2) Find the partial derivative ∂z/∂x of the function:

(z+1)^3y-(zx)^2-3=x/z-ln(xyz)-5xy^2

I have tried some implicit differentiation here, but I can't really make it work out.

Please explain how to solve the two problems very explicitly.

So here you have
$$(z+1)^{3y}-(zx)^2-3=\frac{x}{z}-\ln(xyz)-5xy^2.$$
Is that correct?

Ackbach said:
So you have
$$m=\ln(qh-2h^2)+2e^{(q-h^2+3)^4}-7.$$
Is this correct?

So here you have
$$(z+1)^{3y}-(zx)^2-3=\frac{x}{z}-\ln(xyz)-5xy^2.$$
Is that correct?

1) Yes :-)

2) Not exactly. Here you have it:

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Ok, so for the first one, you have
\begin{align*}
\pd{m}{q}&=\pd{}{q}\left[\ln(qh-2h^2)+2e^{(q-h^2+3)^4}-7\right] \\
&=\frac{1}{qh-2h^2} \, \pd{}{q}(qh-2h^2)+2e^{(q-h^2+3)^4}\,\pd{}{q}\left[(q-h^2+3)^4\right].
\end{align*}
Can you continue? All the normal rules apply: products, quotients, and composition.

For the second one, we use the same procedure we use for functions of a single variable. Here we are assuming that $z=z(x,y)$, and hence we compute:
$$\pd{}{x}\left[ (z+1)^3 y-(zx)^2-3=\frac{x}{z}-\ln(xyz)-5xy^2 \right],$$
or
$$3(z+1)^2y\pd{z}{x}-(2zx^2\pd{z}{x}+2z^2x)=\frac{z-x\pd{z}{x}}{z^2}-\frac{1}{xyz}\pd{(xyz)}{x}-5y^2.$$
Can you continue?

## 1. What are partial derivatives?

Partial derivatives are a type of derivative in multivariable calculus that measures the rate of change of a function with respect to one of its variables, while holding all other variables constant. They are useful for solving problems involving functions with multiple variables.

## 2. How do I solve difficult problems involving partial derivatives?

To solve difficult problems involving partial derivatives, you will need to use various techniques such as the chain rule, product rule, and quotient rule. It is also important to understand the properties of partial derivatives and how to apply them correctly in different scenarios. Practicing and familiarizing yourself with different types of problems will also help improve your problem-solving skills.

## 3. What are the applications of partial derivatives?

Partial derivatives have numerous applications in fields such as physics, engineering, economics, and statistics. They are used to solve optimization problems, determine critical points, and calculate rates of change in systems with multiple variables. They are also essential in understanding the behavior of functions in higher dimensions.

## 4. Can partial derivatives be computed for any type of function?

Yes, partial derivatives can be computed for any type of function, as long as the function has multiple variables. However, the process of computing the partial derivatives may vary depending on the complexity of the function and the techniques needed to solve it.

## 5. Are there any common mistakes to avoid when solving problems involving partial derivatives?

Yes, some common mistakes to avoid when solving problems involving partial derivatives include incorrect application of the rules, forgetting to consider all variables as constants except the one being differentiated, and using the wrong notation. It is important to carefully review your work and double-check your calculations to avoid these errors.

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