Differentiate each of the following functions

Question

differentiate each function

a. f(x)=1/x

b. g(x)=cos^-1 (x^2)

c. y=csc(x)

d. h(x)=ln(lnx)

e. y=e^x* tan*(x^3)

Answer 1

a. First rewrite 1/x = x^-1. Then by the power rule, d/dx x^-1 = -1 * x^-2 = -1/(x²)

b. For this problem we have to know that d/dx arccos(x) = -1/√(1-x²)

We will use the chain rule: d/dx arccos(x²) = -1/√(1-(x²)²) * d/dx x²

Notice that we plugged in x² into the formula for d/dx arccos(x). Also, the chain rule states that we multiply by the derivative of the inner function too, which is d/dx x² = 2x.

Simplifying everything: -2x/√(1-x⁴)

c. Remember that csc(x) = 1/sin(x). Now we can use the chain rule, since we already know the derivatives of 1/x and sin(x). You could also use the quotient rule but I find it is prone to mistakes and in this case the chain rule is simpler.

d/dx 1/sin(x) = -1/sin²(x) * d/dx sin(x) = -cos(x) / sin²(x).

I prefer this form though it is equivalent to -cos(x)/sin(x) * 1/sin(x) = -cot(x) * csc(x)

d. For this problem we have to know that d/dx ln(x) = 1/x. Now we can use the chain rule:

d/dx ln(ln(x)) = 1/ln(x) * d/dx ln(x) = 1/ln(x) * 1/x = 1/(x * ln(x))

e. We will use the product rule and the chain rule. We also need to remember that d/dx tan(x) = sec²(x).

dy/dx = d/dx e^x * tan(x³) + e^x * d/dx tan(x³)

Since e^x is its own derivative, the first term is just e^x * tan(x³)

The second term is more complicated so we use the chain rule. d/dx tan(x³) = sec²(x³) * d/dx x³ = sec²(x³) * 3x². Keep in mind that this term also has an e^x.

Put it all together:

dy/dx = e^x * tan(x³) + sec²(x³) * 3x² * e^x

In these five problems we used the power rule, chain rule, and product rule. We also saw a number of derivatives that are good to know:

d/dx arccos(x) = -1/√(1-x²)

d/dx ln(x) = 1/x

d/dx tan(x) = sec²(x)