How do you find the gradient of a function at a given point?

Answer 1

A function's gradient is also referred to as its slope, and its derivative is the slope (of a tangent) at a specific point on the function.

To find the gradient, take the derivative of the function with respect to #x#, then substitute the x-coordinate of the point of interest in for the #x# values in the derivative.
For example, if you want to know the gradient of the function #y = 4x^3 - 2x^2 +7# at the point #(1, 9)# we would do the following:
Take the derivative with respect to #x#: #12x^2 - 4x#
Substitute the x-coordinate #(x=1)# in for #x#: gradient = #12(1)^2 - 4(1) = 8#
So the gradient of the function at the point #(1, 9)# is #8#.
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Answer 2

To find the gradient of a function at a given point, you need to calculate the derivative of the function with respect to the independent variable. The derivative represents the rate of change of the function at any given point. To find the derivative, you can use differentiation rules such as the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the function. Once you have the derivative, substitute the given point into the derivative expression to obtain the gradient at that point.

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Answer 3

To find the gradient of a function at a given point, you need to calculate the derivative of the function with respect to the variable of interest and then evaluate the derivative at the given point.

  1. Find the derivative: Use differentiation rules (such as the power rule, product rule, quotient rule, and chain rule) to find the derivative of the function with respect to the variable.

  2. Evaluate at the given point: Substitute the coordinates of the given point into the derivative expression to find the gradient (slope) at that point.

  3. Interpretation: The resulting value represents the slope of the tangent line to the curve at the given point.

If the function is given in the form ( y = f(x) ), you'll find the derivative ( f'(x) ) and then evaluate it at the given ( x )-coordinate to find the slope at that point. If the function is given parametrically or implicitly, you'll need to use the appropriate techniques to find the derivative with respect to the variable and then evaluate it at the given point.

Overall, the gradient of a function at a given point represents the rate of change of the function at that specific location, indicating how steeply the function is increasing or decreasing at that point.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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