Please provide the solution on some paper or scanning it easier to follow please and thank you. If not able to please still help cause I am lost on this and this is my final attempt. 

in Calculus Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

I’ll give the answer as clearly as I can. I think it will be a long answer.

Going from past experience with a similar question, I’ll write z in terms of t and s by substitution of x and y. Limitations of the system make it impractical to use photoscans because uploads are restricted, and editing becomes a nightmare as it gets progressively more sluggish. There’s also a restriction on the amount of text in an answer.





Take the middle term:


Rewrite it:


We now use trig and hyp trig identities to simplify the expression:

sinh(10t)≡2sinh(5t)cosh(5t), so sinh(5t)cosh(5t)≡½sinh(10t).

sin(6s)≡2sin(3s)cos(3s), so sin(3s)cos(3s)≡½sin(6s).

(I use ≡ to signify an identity—always true.)

The middle term becomes 48(½sinh(10t)(½sin(6s)).

This simplifies to 12sinh(10t)sin(6s).



∂z/∂t can now be found using the chain rule. For partial differentiation, we treat s as a constant:


I’ve put the constants inside square brackets for clarity. Because we’ve separated the variables t and s we can use d/dt instead of ∂/∂t.

To differentiate we can first write u=cosh(5t) and p=u². du/dt=5sinh(5t) and dp/du=2u.




But sinh(5t)cosh(5t)≡½sinh(10t), so:


And d(sinh(10t))/dt=10cosh(10t).

Now, cosh²(5t)-sinh²(5t)≡1 so sinh²(5t)≡1+cosh²(5t), so the derivatives with respect to t are the same, that is, they both equal 5sinh(10t).


We can now piece together ∂z/∂t:


We need to evaluate this at (t,s)=(ln(2)/10,π/3).

3s=π so sin(3s)=sin(π)=0, sin(6s)=sin(2π)=0, and cos(3s)=cos(π)=-1. Any terms containing sin(3s) or sin(6s) will be zero and can be ignored, so at the given (t,s):


But 10t=ln(2). Sinh(ln(2))=½(e^ln(2)-e^-ln(2))=½(2-½)=½(3/2)=¾.




Let u=sin(3s), du/ds=3cos(3s), p=u², dp/du=2u.



so differentiating both sides: d(cos²(3s))=-d(sin²(3s))=-3sin(6s).



At the given (t,s):

So, because the first and last terms are zero (sin(3s)=sin(6s)=0):


by Top Rated User (1.1m points)

Related questions

1 answer
asked Feb 19, 2020 in Calculus Answers by anonymous | 199 views
1 answer
asked Feb 19, 2020 in Calculus Answers by anonymous | 451 views
1 answer
asked Apr 8, 2020 in Calculus Answers by qwertykl Level 2 User (1.4k points) | 712 views
1 answer
1 answer
1 answer
asked Apr 24, 2013 in Calculus Answers by anonymous | 627 views
1 answer
asked Mar 26, 2013 in Calculus Answers by anonymous | 543 views
1 answer
asked Nov 8, 2012 in Calculus Answers by anonymous | 416 views
0 answers
asked Mar 12, 2013 in Calculus Answers by anonymous | 440 views
Welcome to, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method least common multiple proving trigonometric identities factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than division fractions geometry how to find x intercept differentiation exponents 8th grade math simplifying fractions equivalent fractions geometry 10th grade inverse function area of a triangle elimination method story problems standard deviation integral simplify systems of equations containing three variables ratios width of a rectangle percentages area of a circle place value parallel lines solving triangles mathematical proofs circumference of a circle solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems number of sides of a polygon quadratic functions length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets simplifying radicals set builder notation significant figures arithmetic sequences median age problem graphing please help me to answer this step by step number patterns adding fractions derivatives radicals trigonometry product of two consecutive numbers roots of polynomials midpoint of a line limits decimals please help pre-algebra problems compound interest divisibility rules graphing functions subtracting fractions numbers discrete mathematics volume of a cylinder simultaneous equations integration angles probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 logarithms how to complete the square matrices mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,261 questions
97,770 answers
16,706 users