Find ∂f / ∂x, ∂f / ∂y if (a) f(x, y)=x y (b) f(x, y)=e^x y (c) f(x, y)=x cosx cosy (d) f(x, y)=(

Find ∂f / ∂x, ∂f / ∂y if (a) f(x, y)=x y (b) f(x, y)=e^x y...

Key Concepts

Partial Derivatives

Partial derivatives represent the rate at which a multivariable function changes with respect to one variable while holding the other variables constant. Understanding partial differentiation is crucial for analyzing how individual variables influence the function's output in multivariable calculus.

Product Rule

The product rule is a technique used to differentiate functions that are the product of two or more functions. When differentiating a product, each function is differentiated in turn while treating the others as constants, and the results are summed appropriately.

Chain Rule

The chain rule is an essential method for differentiating composite functions, where one function is nested inside another. It states that the derivative of the composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Derivative of Exponential Functions

When differentiating exponential functions, such as those involving e raised to a function, the derivative is given by the original exponential function multiplied by the derivative of the exponent. This rule is a direct application of the chain rule.

Derivative of Logarithmic Functions

Differentiating logarithmic functions involves applying the rule that the derivative of the logarithm of a function is the derivative of that function divided by the original function, often coupled with the chain rule when the argument of the logarithm is itself a composite function.

Trigonometric Derivatives

Trigonometric derivatives refer to the rules used to differentiate functions involving trigonometric functions like sine and cosine. Familiarity with these derivatives, including the derivatives of cosine and sine, is necessary for handling functions where trigonometric operations are combined with other mathematical operations.

Transcript

00:01 So we'd like to find the partial x and y derivatives of a couple of functions first of all a so we have fx that is the partial f derivative partial x derivative partial derivative with respect to x it's just well y is a constant with respect to x so d dx y times x is y and symmetrically the y derivative is just x i get that the way we do that of course is by just taking x derivative we treat y as a constant and differentiate with respect to x.

00:35 This is a linear polynomial in x so we just get y to differentiate with respect to y we treat x as a constant this is linear in y and the derivative is the slope which is x for b here again, we treat y as a constant fx will be let's see by the chain rule the derivative of the inner part is just y times e to the xy is the derivative of the this whole thing e to the xy times d dx x times y which is y and then by symmetry again fy is x times e to the xy for c here x times cosine x cosine y i'm actually gonna do the y derivative first because x times cosine x all of this is just constant with respect to y and we get x cosine x sine y and of course this negative sine y for the x derivative we're actually gonna have to do the product rule though so f sub x is going to be cosine y is just a constant and then the x derivative of x times cosine x which is cosine x minus x times sine x by the product rule derivative of x times cosine x is just cosine x x times the derivative of cosine x is negative x times sine x and finally here our d is going to be let's see here fx is we need the product rule again derivative of this thing here is just 2x we got log of x squared plus y squared plus this part here x squared plus y squared times the derivative of all this that's going to be derivative of log is 1 over x squared plus y squared or derivative of log is 1 over x so we just do 1 over that this all cancels out and we just take the derivative with left here is the derivative of the inner part which is 2x all right to expand that out real quick this it will be derivative of this times...

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