Step 1 - Differentiate the function using the chain rule. We can do this by letting . We can therefore write the derivative as:

The gradient function can be further simplified to:

Step 2 - Substitute in into the equation:

We will use the chain rule to answer this question.

Use the fact that where u is a function of , to differentiate the function.

Simplifying this will give:

Step 1 - The logarithmic functions have base 3 and 5 respectively, meaning we are unable to differentiate them directly. Thus, we must manipulate the function to change our logs to base .

We can accomplish this using the change of base law which states .

This can be further simplified to:

Step 2 - We can now use the chain rule to differentiate the function. Note: the denominator is just a constant so it can be taken out of the derivative.

Which can be simplified to give you the final answer:

First use the log subtraction law to express as:

Then differentiate:

This allows us to use basic differentiation methods instead of attempting a chain + quotient rule.

First use the log power down law to simplify the expression;

Then differentiate;

This question can be made much easier by simplifying the logarithmic expression using the log laws.

Step 1 - Apply the log laws to simplify the expression

We begin by changing the square root inside the log to a power of half:

We can begin by applying the logarithmic power down law :

We can use the fact that to simplify the equation further:

This removes the need to utilise the quotient rule when differentiating the function.

Step 2 - Finally, differentiate the function, utilising the chain rule and the fact that , where u is a function of :

This will simplify to:

Finding the common denominator, we can further simplify this to:

Which once simplified, will give you the final answer:

This can first be simplified using the log addition law as seen below:

Next, differentiate each logarithm separately;

And use the chain rule where necessary:

Step 1 - Rewrite and simplify the expression, using the logarithmic power down law which states . This will make the expression easier to differentiate:

Step 2 - To differentiate this, we must change the logarithm to base . This can be done using the change of base law; .

Step 3 - Use the product rule to differentiate the expression above ( Note the denominator is just a constant so it can be moved to the front):

This can be further simplified to:

Substituting an out results in the final answer being:

Step 1 - First, use the log quotient rule to seperate the equation:

Step 2 - Repeatedly apply the log product rule to further simplify the equation:

Step 3 - Evaluate and use the power rule to simplify

Step 4 - Differentiate with respect to :

Therefore the derivative is

Step 1 - Find the derivative of using the quotient rule. Let and This therefore gives and . We can then apply the quotient rule as seen below:

Simplifying this, we get:

Step 2 - Substitute the value , to solve for the gradient of the tangent at that point:

Step 3 - Finding the equation of the tangent using the point gradient formula.

And now converting to general form:

Part A

Step 1 - Differentiate C with respect to . Note, the constant stays at the front of the derivative.

Step 2 - Substitute and :

Part B

In order to find in terms of , simply substitute into as shown:

We can simplify this to:

Part C

Step 1 - In order to differentiate in terms of t, we first need to use the change of base law: as follows:

Step 2 - Differentiate in terms of using the chain rule:

notice that

Step 3 - Substitute the values and into as follows:

at :

at t = 10:

Part D

From part C step 2, we can see that the rate of change of is in the numerator for . Therefore, the rate of change of is directly

proportional to .