AI tutor
Welcome to Bytelearn!
Let’s check out your problem:
Given that , find in terms of only .Answer:
Full solution
Q. Given that , find in terms of only .Answer:
- Expand : First, we need to express in terms of using the given expression for . We have . To find , we need to raise the expression for to the fourth power.
- Calculate : We calculate by raising to the fourth power. This involves expanding the binomial using the binomial theorem or by multiplying the expression by itself four times. However, since we are ultimately interested in the derivative with respect to , we can simplify our work by differentiating directly without fully expanding the binomial.
- Chain rule for : The derivative of with respect to can be found using the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In this case, the outer function is (where ) and the inner function is .
- Combine derivatives: Using the chain rule, we differentiate with respect to and then multiply by the derivative of with respect to . The derivative of with respect to is . The derivative of with respect to is the derivative of , which is .
- Differentiate : Now we combine the two derivatives to find the derivative of with respect to : .
- Sum of derivatives: Next, we need to differentiate with respect to . The derivative of with respect to is .
- Substitute into derivative: We now have the derivatives of both terms in the expression . The derivative of the entire expression with respect to is the sum of the derivatives of the individual terms: .
- Substitute into derivative: We now have the derivatives of both terms in the expression . The derivative of the entire expression with respect to is the sum of the derivatives of the individual terms: .Finally, we substitute into the expression for the derivative to express it entirely in terms of : .
