Example Question - power rule differentiation
Here are examples of questions we've helped users solve.
Derivative of a Function with Simplification
The function \( h(x) \) is given as: \[ h(x) = \frac{4x^3 - 7x + 8}{x} \] Before we find the derivative, we can simplify the function by dividing each term in the numerator by \( x \). This gives us: \[ h(x) = 4x^2 - 7 + \frac{8}{x} \] Now, let's find the derivative of \( h(x) \) with respect to \( x \) using the power rule and the derivative of a constant. The power rule states that the derivative of \( x^n \) with respect to \( x \) is \( n \cdot x^{n-1} \), and the derivative of a constant is 0. For the term \( \frac{8}{x} \), which can be written as \( 8x^{-1} \), we also use the power rule. Let's differentiate each term: 1. The derivative of \( 4x^2 \) with respect to \( x \) is \( 2 \cdot 4x = 8x \). 2. The derivative of \( -7 \) (a constant) with respect to \( x \) is \( 0 \). 3. The derivative of \( \frac{8}{x} \) (which is \( 8x^{-1} \)) with respect to \( x \) is \( -1 \cdot 8x^{-1 - 1} = -8x^{-2} \). Putting it all together, the derivative of \( h(x) \) is: \[ h'(x) = 8x - 0 - \frac{8}{x^2} \] \[ h'(x) = 8x - \frac{8}{x^2} \] So, the derivative of the function \( h(x) \) is \( 8x - \frac{8}{x^2} \).
Derivative Calculation of a Quadratic Function
The image shows a mathematical problem where you are given a function f(x) = 3x^2 + 2x and asked to find the derivative of the function at x = 2, represented as f'(2). To find the derivative f'(x) of the function f(x) = 3x^2 + 2x, we will use the power rule for differentiation. The power rule states that the derivative of x^n is n*x^(n-1). So for f(x) = 3x^2 + 2x: f'(x) = d/dx (3x^2) + d/dx (2x) = 3 * 2x^(2-1) + 2 * 1x^(1-1) = 6x + 2 Now we need to evaluate the derivative at x = 2: f'(2) = 6(2) + 2 = 12 + 2 = 14 Therefore, the derivative of the function at x = 2, f'(2), is 14.
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