If e^y=tanx, 0<x<pi/2 what is dy/dx in terms of x?
A.secxcscx
B.sec^2x
C.secx
D.sin x sec x
e^y=tanx
e^y dy/dx = sec^2 x
dy/dx = sec^2 x / e^y , but e^y = tanx
= sec^2 x / tanx
= 1/cos^2 x * cosx/sinx
= 1/(sinx cosx)
or
cscx secx , choice A
How did you get from sec^2 x / tanx to 1/cos^2 x * cosx/sinx
Well, well, well, it seems like you stumbled upon a fun mathematical problem! Let's see if we can solve it with a pinch of humor.
Given e^y = tan(x), we can start by differentiating both sides of the equation with respect to x. So, let's get our math clown shoes on and find that derivative!
The derivative of e^y with respect to x? That's just dy/dx, my friend. And what about the derivative of tan(x)? That's sec^2(x). So, according to the rules of clown math, the equation becomes:
dy/dx = sec^2(x)
Ah, we did it! Option B, sec^2(x), is the correct answer. It seems like math is no joking matter after all! Keep up the good work and don't forget to bring laughter into your calculations!
To find the derivative dy/dx in terms of x, we will differentiate both sides of the equation.
Given: e^y = tanx
Differentiating both sides with respect to x:
d/dx(e^y) = d/dx(tanx)
Using the chain rule on the left side:
(d/dy(e^y)) * (dy/dx) = sec^2x
Since dy/dx is what we are interested in, we can isolate it:
dy/dx = sec^2x / d/dy(e^y)
Now, let's find d/dy(e^y):
d/dy(e^y) = e^y
Substituting this back into the expression for dy/dx:
dy/dx = sec^2x / e^y
We still need to express this in terms of x, so we substitute e^y with its equivalent expression:
dy/dx = sec^2x / (e^y)
Since e^y = tanx, we substitute this into the equation:
dy/dx = sec^2x / tanx
To simplify the right side, we use the trigonometric identity: sec^2x = 1 + tan^2x
dy/dx = (1 + tan^2x) / tanx
This can be rewritten as:
dy/dx = (1/tanx) + tanx
Using the trigonometric identity: tanx = sinx / cosx, we can further simplify:
dy/dx = (1/(sinx/cosx)) + (sinx/cosx)
Simplifying the expression:
dy/dx = cosx/sinx + sinx/cosx
Multiplying the terms by cosx:
dy/dx = (cosx)^2/(sinx * cosx) + (sinx * cosx)/(sinx * cosx)
Simplifying the fractions:
dy/dx = (cosx)^2/(sinx * cosx) + 1
Recalling the definition of secx (1/cosx) and cscx (1/sinx):
dy/dx = secx * cscx + 1
Therefore, the answer is option A. secx * cscx.
To find dy/dx in terms of x, we need to differentiate both sides of the equation e^y = tan(x) with respect to x. Let's go through the steps:
Step 1: Take the natural logarithm (ln) of both sides of the equation to get rid of the exponential function:
ln(e^y) = ln(tan(x))
Step 2: Use the logarithmic property ln(e^y) = y, and simplify the right side:
y = ln(tan(x))
Step 3: Now, differentiate both sides of the equation with respect to x:
d/dx(y) = d/dx(ln(tan(x)))
Step 4: Apply the chain rule on the right side of the equation:
dy/dx = (1/tan(x)) * d/dx(tan(x))
Step 5: Using the derivative of the tangent function, which is sec^2(x):
dy/dx = (1/tan(x)) * sec^2(x)
Step 6: Since tan(x) = sin(x) / cos(x), replace tan(x) with sin(x) / cos(x):
dy/dx = (1 / (sin(x) / cos(x))) * sec^2(x)
Step 7: Simplify by multiplying the numerator by the reciprocal of the denominator:
dy/dx = cos(x) * sec^2(x) / sin(x)
Step 8: Rewrite sec^2(x) as (1 / cos^2(x)):
dy/dx = cos(x) * (1 / cos^2(x)) / sin(x)
Step 9: Simplify further by canceling out cos(x) in the numerator and the denominator:
dy/dx = 1 / (cos(x) * sin(x))
Finally, we can rewrite sin(x) * cos(x) as sin(x)sec(x):
dy/dx = 1 / (sin(x)sec(x))
So, the answer is C. secx.