Derivatives Question

Question

Find the derivative of f(x)=sinx/√x

Answer 1

Hi Shade,

to find this derivative, we'll need to use some derivative identities and the quotient rule.

The derivative of sin x is best found by simply memorizing its derivative. Here's all 6 I recommend you memorize:

sin(x) -> cos(x)

cos(x) -> -sin(x)

tan(x) -> sec²(x)

sec(x) -> sec(x)tan(x)

csc(x) -> -csc(x)cot(x)

cot(x) -> -csc²(x)

tip: any trig function that starts with "co" like cosine, cosecant and cotangent will have its derivative be negative, as shown.

That being said, we can now apply the quotient rule that goes as follows:

((low*diHi) - (hi*diLow)) / (low * low)

low = the function in the denominator

diLow = derivative of the function in the denominator

hi = the function in the numerator

diHi = the derivative of the function in the numerator.

We found the derivative of the function in the numerator (sin(x)) earlier from the list provided (I highly recommend you memorize them) so we need the derivative of the function in the denominator (√x).

that can be rewritten as x^1/2 and now we can apply the power rule (nx^n-1) and we end up with:

1/2*(x^-1/2) or 1/(2√x)).

Now we can plug in all the pieces together and end up with:

(√(x)*cos(x)) - (sin(x) * 1/(2√x))) / (√x * √x)

For some teachers/professors, this is enough, but you can always simplify it further like having the (√x * √x) cancel out and give us (x) in the denominator. I'll stop here but feel free to continue simplifying.

f'(x) = (√(x)*cos(x)) - (sin(x) * 1/(2√x))) / (√x * √x)