How to Differentiate z with Respect to ∅?

To find this derivative, we use the power rule and chain rule. First, we differentiate ((A+B)/2) with respect to ∅, which gives us 0 since A and B are constants. Then, we differentiate ((A-B)/2)Cos2∅ with respect to ∅, using the chain rule to get -(A-B)sin(2∅). Finally, we differentiate C Sin2∅ with respect to ∅, again using the chain rule to get 2Ccos(2∅). This gives us the full derivative dz/d∅. To solve for tan(2∅), we set dz/d∅ equal to 0
  • #1
I am having trouble Differentiating the following with respect to ∅. Can you show me the steps I need to take to get the given correct answer please.

z = ((A+B)/2) + ((A-B)/2)Cos2∅ + C Sin2∅


The correct answer is
Tan2∅ = (2C)/(A-B)
 
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  • #2
for your answer to be true, dz/d(theta) has to equal zero.

differentiating: dz/d(theta)=0-(A-B)sin(2∅)+2Ccos(2∅)
if dz/d∅=0, then divide by cos(2∅) on both sides and you'l get

0=-(A-B)tan(2∅)+2C

solve for tan and you'll get tan(2∅)=(2C)/(A-B)
 
  • #3
Thanks Physeven
can you explain to me in more detail how you got the line
please. Its the actual process of differentiation I am having trouble with.

dz/d(theta)=0-(A-B)sin(2∅)+2Ccos(2∅)
 

What is a differential equation?

A differential equation is an equation that involves an unknown function and its derivatives. It is used to describe the relationship between a function and its rate of change, and is commonly used in mathematical and scientific models.

What is the difference between an ordinary and partial differential equation?

An ordinary differential equation (ODE) involves a single independent variable, while a partial differential equation (PDE) involves multiple independent variables. ODEs are used to describe phenomena that change over a single dimension, while PDEs are used to describe phenomena that change over multiple dimensions.

What are some real-world applications of differential equations?

Differential equations have numerous applications in fields such as physics, engineering, economics, and biology. They are used to model the motion of objects, the flow of fluids, the growth of populations, and many other dynamic processes.

How are differential equations solved?

The process of solving a differential equation involves finding a function that satisfies the equation. This can be done analytically, using mathematical techniques and formulas, or numerically, using computer algorithms to approximate the solution.

Why are differential equations important in science?

Differential equations are important in science because they allow us to mathematically describe and predict the behavior of complex systems and phenomena. They provide a powerful tool for understanding the relationships between different variables and how they change over time.

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