Partial Differentiation - Total Derivatives

Category: Differential Calculus

"Published in Newark, California, USA"

Find ^du/_dt for the following functions:

Solution:

Consider the given equation above

Since the first function has three variables, then we have to take the partial derivatives of a function with respect to x and y as follows

Next, take the derivative of the other functions with respect to x since they have only one variables as follows

Therefore,

Substitute the values of x and y in the above equation to obtain the final equation in terms of u and t as follows

Differentiation - Rate Problem, 2

Category: Differential Calculus, Trigonometry

"Published in Newark, California, USA"

If an angle θ increases uniformly, find the smallest positive value of θ for which tan θ increases 8 times as fast as sin θ. 

Solution:

The given word problem is about the rate problem of an angle θ and its trigonometric functions. 

From the word statement, "...for which tan θ increases 8 times as fast as sin θ." then the working equation will be

Take the derivative on both sides of the equation with respect to time t as follows

As you notice that we can cancel the angular rate which is ^dθ/_dt on both sides of the equation and we can solve for the value of angle θ as follows

Take the cube root on both sides on the equation, we have

Take the inverse cosine on both sides of the equation,

Therefore,

Proving Trigonometric Identities, 5

Category: Trigonometry

"Published in Newark, California, USA"

Prove that 

Solution:

Consider the give equation above

Use the left side of the equation to prove the trigonometric identities because it is more complicated part, as follows

Use the sum and product formula, we have

Multiply the numerator and the denominator by sin x, we have

but

Therefore the above equation becomes

Therefore,