How do I use the product rule to solve these two questions. Much appreciated! xo
Find the derivative for k(x) = (5x4 + 2)(3sin x)
Find the derivative for m(x) = -ex cos x at x = 1
First familarize yourself with the "product rule" for derivatives.
For a product of two functions, which I will call
f(x) = u(x) * v(x),
the derivative of the product function is
df/dx = u dv/dx + v du/dx
For your second question, let
u(x) = -e^x and v(x) = cos x
du/dx = -e^x and dv/dx = -sin x
df/dx = -e^x cos x + e^x sin x
Now you try the other one
To use the product rule, which states that the derivative of the product of two functions is equal to the first function times the derivative of the second function plus the second function times the derivative of the first function, follow these steps:
1. Identify the two functions within the given equation.
2. Differentiate each function separately.
3. Apply the product rule by multiplying the first function with the derivative of the second function and adding it to the product of the second function with the derivative of the first function.
Let's go through the process for each question:
Question 1: Find the derivative for k(x) = (5x^4 + 2)(3sin x)
Step 1: Identify the two functions:
First function: 5x^4 + 2
Second function: 3sin x
Step 2: Differentiate each function:
The derivative of the first function is obtained by applying the power rule. We bring down the exponent and multiply it by the coefficient:
Derivative of the first function = 20x^3
The derivative of the second function, sin x, is obtained by applying the chain rule. The derivative of sin x is cos x, and since it is a function of x, there is no need to multiply it by the chain rule term.
Step 3: Apply the product rule:
Derivative of k(x) = (5x^4 + 2)(cos x) + (3sin x)(20x^3)
= 5x^4 cos x + 2cos x + 60x^3 sin x
Question 2: Find the derivative for m(x) = -ex cos x at x = 1
Step 1: Identify the two functions:
First function: -ex
Second function: cos x
Step 2: Differentiate each function:
The derivative of the first function, -ex, is -ex itself since the derivative of e^u is e^u multiplied by the derivative of u.
The derivative of the second function, cos x, is -sin x.
Step 3: Apply the product rule:
Derivative of m(x) = (-ex)(-sin x) + (cos x)(-ex)
At x = 1, plug in the value to find the derivative:
Derivative of m(x) at x = 1 = (-e^1)(-sin 1) + (cos 1)(-e^1)
= esin 1 - ecos 1
This is the derivative of m(x) at x = 1.
Remember to always check for simplifications and to apply the rules of differentiation correctly. I hope this helps!