Differentiate -7xy³
To differentiate -7xy³, we can differentiate each term separately using the power rule.
The power rule states that if we have a term of the form cx^n, where c is a constant and n is a variable with a constant coefficient, we can differentiate it by multiplying the coefficient by the exponent and then subtracting 1 from the exponent.
Let's differentiate each term separately:
Differentiating -7:
The constant -7 does not have any variables, so its derivative is 0.
Differentiating x:
The variable x has an exponent of 1, so its derivative is 1.
Differentiating y³:
The variable y has an exponent of 3, so its derivative is 3y^2.
Putting it all together, the derivative of -7xy³ is:
0x + 1(-7)y³ + (-7x)3y²
Simplifying this expression, we get:
-7y³ - 21xy²
Therefore, the derivative of -7xy³ is -7y³ - 21xy².
hmmm
Differentiate -7xy³
say p = -7 x y^3
then
dp = d/dx [ -7 x y^3] dx + d/dy [ -7 x y^3 ] dy
so
dp = [ -7 y^3 ] dx + [-21 x y^2 ] dy
Yes, that's correct! When differentiating with respect to multiple variables, we can use the partial derivative notation to indicate which variable we are differentiating with respect to. In this case, we have p = -7xy³ and we want to find dp.
Using the partial derivative notation, we have:
dp = (∂p/∂x) dx + (∂p/∂y) dy
Taking the partial derivatives, we find:
∂p/∂x = -7y³ (since ∂x/∂x = 1 and ∂y/∂x = 0)
∂p/∂y = -21xy² (since ∂x/∂y = 0 and ∂y/∂y = 1)
Substituting these results into the previous equation, we obtain:
dp = (-7y³) dx + (-21xy²) dy
So, dp = (-7y³) dx + (-21xy²) dy.
To differentiate -7xy³, we will apply the power rule for differentiation. The power rule states that if we have a function of the form f(x) = ax^n, where "a" is a constant and "n" is a real number, the derivative of f(x) is given by:
f'(x) = nax^(n-1).
Applying this rule to -7xy³, we have:
(-7xy³)' = -7(3)(x)^(3-1)(y³)'.
Simplifying further:
(-7xy³)' = -21x²(y³)'.
Since the derivative of y³ with respect to x is zero (because y³ does not contain x), we have:
(-7xy³)' = -21x²(0).
Simplifying this expression, we get:
(-7xy³)' = 0.
Therefore, the derivative of -7xy³ with respect to x is zero.
To differentiate the algebraic expression -7xy³, we need to apply the power rule of differentiation. The power rule states that when differentiating a term, we multiply the term by its exponent and then decrease the exponent by 1.
Let's break down the given expression step by step:
1. Start with the constant term (-7): Since it does not contain any variables, the derivative of -7 is 0.
2. Differentiate the term "x": The coefficient of "x" is 1 (since it is not explicitly written), and the exponent is 1. Applying the power rule, we multiply 1 by 1 and decrease the exponent by 1 to get 1x¹. Therefore, the derivative of x is simply 1.
3. Differentiate the term "y³": Similar to the previous step, we multiply the coefficient 1 (since it is not explicitly written) by the exponent 3 to get 1y³. We then decrease the exponent by 1 to get y². Thus, the derivative of y³ is 3y².
Putting it all together, the derivative of -7xy³ is:
0 (constant term) + 1x¹ (differentiated x) + 3y² (differentiated y³)
Therefore, the final result is 1x + 3y², or simply x + 3y².