# Differntiation Bloody Confusing

• dagg3r
In summary: The '5' comes from the chain rule.In summary, in the first problem, the person is asking for help with finding the derivative of a function involving t and using the chain rule or possibly the logarithmic method. In the second problem, the person is trying to find the rate of change between two moving cars using the distance formula and possibly implicit differentiation. They have also tried using product rule, but are unsure if they have done it correctly.
dagg3r
Differntiation! Bloody Confusing!

HI guys help me out here with some differentitaion problems i'll post here what i have done and show me the way to go thanks
1. find dx/dt where x=(4-t)^5t

i tried using the chain rule but it doesn't work though cos you got that t constant do i use the logarithic way of differntiation so log X=5t*log(4-t) then i diff this?

2. A red car is traveling east towards an intersection at a speed of 80km/hr while a blue car simultaneously traveling north away from the intersection at a speed of 60 km/hr. If the red car is 4km from the intersection and the blue car is 3km from the intersection what is the rate of change the cars are changing?

i drew pictures of this then started to think to try and use pythagoras and possibly use some differentiation there applying the direction changes of negatives and positive but got lost and somebody point me the steps on to solve this problem thanks

1. Try implicit differentiation
2. Set up a system of equations. I assume you meant RoC of the distance between the cars. The distance formula will probably come in handy.

Eq of motion for the cars are
x1 = 80t + 4
x2 = 60t - 3

The distance btwn the two functions as a function of time is the distance between the coordinates of x1 and x2 at any given time.

hmm how can you do implicit differentiation when its to the power of 5t? i tried to the log way and got this

lnx=5tloge(4-t)
then used product rule with u=5t v=loge(4-t)

dy/dx=(4-t)^5t * [ 5t(-1/4-t) + 5ln(4-t) ]

realling long and ugly i think i did it wrong reckon somebody can show me how to apply
implicit differentiation usually i can do it but got confused with the power of 5t

dagg3r said:
hmm how can you do implicit differentiation when its to the power of 5t? i tried to the log way and got this

lnx=5tloge(4-t)
then used product rule with u=5t v=loge(4-t)

dy/dx=(4-t)^5t * [ 5t(-1/4-t) + 5ln(4-t) ]

realling long and ugly i think i did it wrong reckon somebody can show me how to apply
implicit differentiation usually i can do it but got confused with the power of 5t
Of course, you DON'T mean "dy/dx"

$$\frac{1}{ln x} \frac{dx}{dt}= 5 ln(4-t)- \frac{5t}{4-t}$$
so
$$\frac{dx}{dt}= (4-t)^{5t}(5 ln(4-t)- \frac{5t}{4-t})$$

looks like just what you have.

HallsofIvy said:
Of course, you DON'T mean "dy/dx"

$$\frac{1}{ln x} \frac{dx}{dt}= 5 ln(4-t)- \frac{5t}{4-t}$$
so
$$\frac{dx}{dt}= (4-t)^{5t}(5 ln(4-t)- \frac{5t}{4-t})$$

looks like just what you have.

And of course you don't mean $$\frac{1}{ln x} \frac{dx}{dt}$$

You mean

$$\frac{d}{dt} ln x =\frac{1}{x} \frac{dx}{dt}= 5 ln(4-t)- \frac{5t}{4-t}$$
so
$$\frac{dx}{dt}= (4-t)^{5t}(5 ln(4-t)- \frac{5t}{4-t})$$

Oops: $$\frac{1}{x}\frac{dx}{dt}$$

## 1. What is differentiation?

Differentiation is a mathematical concept that involves finding the rate at which one variable changes with respect to another variable. It is often used in calculus to solve problems involving rates of change and optimization.

## 2. What is the difference between differentiation and integration?

While differentiation involves finding the rate of change of a function, integration involves finding the area under a curve. In other words, differentiation is the opposite process of integration.

## 3. How is differentiation used in real life?

Differentiation has many real-life applications, such as in physics to calculate velocity and acceleration, in economics to determine marginal cost and revenue, and in biology to model population growth and decay.

## 4. What are the basic rules of differentiation?

The basic rules of differentiation include the power rule, product rule, quotient rule, and chain rule. These rules can be used to find the derivative of a wide variety of functions.

## 5. Is differentiation always necessary?

No, differentiation is not always necessary. In some cases, it may be more efficient to solve a problem using other methods such as algebra or geometry. However, differentiation is a powerful tool that can greatly simplify the solution process for many problems.

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