Question

Find the derivative with respect to x of the quantity e raised to the power of 4 times the natural log of x.

a) 4e⁴ˡⁿ(ˣ)
b) (4e⁴ˡⁿ(ˣ))/x
c) (4e⁴ˡⁿ(ˣ)))/(ˣˡⁿ(¹⁰))
d) 4e⁴ˡⁿ(ˣ))ˡⁿ(¹⁰)

Answer 1

The correct derivative of the function e raised to the power of 4 times the natural log of x with respect to x is (4e^(4lnx))/x, using the chain rule of differentiation and properties of logarithms and exponentiation. The correct derivative of the function with respect to x is (4e^(4lnx))/x, which corresponds to option (b).

Step-by-step explanation:

The question asks to find the derivative of the function e raised to the power of 4 times the natural log of x (written as e^(4lnx)). To solve this, we use the chain rule of differentiation and a property of logarithms which states that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number. This simplifies our expression to e^(ln(x^4)), which is just x^4 because e and the natural log are inverse functions.

The derivative of x^4 with respect to x is 4x^3. However, since we began with e^(4lnx), we actually need to chain the derivative of the inner function 4lnx, which is 4/x. Multiplying these derivatives together, we get the final derivative: (4x^3)(4/x) = 4e^(4lnx)/x.

Hence, the correct derivative of the function with respect to x is (4e^(4lnx))/x, which corresponds to option (b).