Example Question - mathematical functions
Here are examples of questions we've helped users solve.
Differentiation of Mathematical Functions
The image is blurry, but it seems to show four mathematical functions that need to be differentiated with respect to \( x \). I'll attempt to identify them and differentiate each one: (a) \( y = (x + n)^1 \) Using the power rule, we get: \( \frac{dy}{dx} = 1 \cdot (x + n)^{1 - 1} \cdot \frac{d}{dx}(x + n) \) \( \frac{dy}{dx} = (x + n)^0 \cdot \frac{d}{dx}(x + n) \) Since \( (x + n)^0 = 1 \) and \( \frac{d}{dx}(x + n) = 1 \), then \( \frac{dy}{dx} = 1 \cdot 1 = 1 \) (b) \( f(x) = (2x + n)^1 \) Applying the power rule: \( f'(x) = 1 \cdot (2x + n)^{1 - 1} \cdot \frac{d}{dx}(2x + n) \) \( f'(x) = (2x + n)^0 \cdot \frac{d}{dx}(2x + n) \) Since \( (2x + n)^0 = 1 \) and \( \frac{d}{dx}(2x + n) = 2 \), then \( f'(x) = 1 \cdot 2 = 2 \) (c) \( y = (3 - 4x)^5 \) Using the chain rule, we get: \( \frac{dy}{dx} = 5(3 - 4x)^{4}(-4) \) \( \frac{dy}{dx} = -20(3 - 4x)^{4} \) (d) \( g(x) = (3z - 4x)^2 \) (There appears to be a typo in the variable used in the original function. Assuming it's supposed to be \( x \), not \( z \), and differentiating accordingly:) \( g'(x) = 2(3 - 4x)^1(-4) \) \( g'(x) = -8(3 - 4x) \) If the variables \( n \) or \( z \) are constants, then my differentiation is correct. If \( n \) or \( z \) are not constants, and you meant a different variable or power, please provide the correct expressions.
Solving for y with Given Data Points
The image contains two parts of information: 1. \( y = 21 \) when \( x = 9 \) 2. \( y = ? \) when \( x = -6 \) To solve the second part of the question, we need to find a relationship between x and y that is consistent with the given data. However, with only one data point given (\( y = 21 \) when \( x = 9 \)), we cannot determine the exact relationship between x and y because there are infinitely many mathematical functions or relationships that could satisfy this single condition. To proceed, we need either a specific function relating x to y, or more data points that would allow us to infer such a relationship. Without this information, we cannot determine the value of y when \( x = -6 \). If you have additional information such as a mathematical equation that relates x to y, please provide it so that the question can be solved.
Calculating a Function Value
The image shows a function definition and a request to evaluate the function at a particular point: f(x) = x^2 - x; f(-4) To solve this, we simply substitute x with -4 in the function's equation: f(-4) = (-4)^2 - (-4) f(-4) = 16 + 4 f(-4) = 20 Therefore, f(-4) is equal to 20.
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