Use implicit differentiation to find the derivative of: Write the solution of each number

Question

10. sech²x + csch²y = 10

ans. y' = tanh³x - tanh x / coth³y - coth y

Answer 1

Differentiation of sech²x + csch²y = 10 gives

d(sech²x)/dx + d(csch²y)/dx = 0

d(sech²x)/dx = 2sech²x d(sechx)/dx = 2sechx(–sechx tanhx) = –2sech²x tanhx

d(csch²y)/dx = 2csch²y d(cschy)/dx = 2cschy(–cschy cothy)y' = –2csch²y cothy y'

which means that

–2sech²x tanhx – 2csch²y cothy y' = 0

which means that

y' = (2sech²x tanhx)/(–2csch²y cothy) = (sech²x tanhx)/(–csch²y cothy)

Hyperbolic Trig Identities

sech²x = 1 – tanh²x

csch²y = coth²y – 1

which means that

y' = (sech²x tanhx)/(–csch²y cothy) = ((1 – tanh²x) tanhx)/(–(coth²y – 1) cothy) =

((1 – tanh²x) tanhx)/((1 – coth²x) cothy) =

(tanhx – tanh³x)/(cothx – coth³x)

which is the same thing as

y' = (tanh³x – tanhx)/(coth³x – cothx)