Compute d w / d t w=(z)/(x y^2)-3 ; x=(1)/(t^2), y=-5 t, z=√(t)

Name: Problem 16, Chapter 13: Calculus with Concepts in Calculus
Uploaded: 2022-04-17T11:44:57-08:00
Duration: 2 min 13 s
Channel: Shivani Yadav
Description: Compute d w / d t w=(z)/(x y^2)-3 ; x=(1)/(t^2), y=-5 t, z=√(t)

step 1 The given functions are W equals to Z divided by xY square negative of 3, x equals to 1 over t s

Compute d w / d t w=(z)/(x y^2)-3 ; x=(1)/(t^2), y=-5 t,...

Key Concepts

Substitution

Substitution involves replacing variables with their explicit expressions in terms of the independent variable, simplifying the process of differentiation. By substituting the definitions of intermediate variables, one can transform a complex, composite expression into a function solely of the independent variable, making the application of the chain, product, and quotient rules more straightforward.

Power Rule

The power rule is a straightforward technique for differentiating functions of the form t^n, where n is any real number. The derivative is obtained by multiplying by the exponent and then reducing the exponent by one (n*t^(n-1)). This rule is widely used in calculus, especially when dealing with algebraic expressions involving powers of the independent variable.

Product Rule

The product rule is used when differentiating the product of two functions. It states that if you have two functions multiplied together, their derivative is the sum of the derivative of the first multiplied by the second plus the first multiplied by the derivative of the second. This rule is particularly important when a function is rewritten as a product of simpler functions to facilitate differentiation.

Quotient Rule

The quotient rule is applied when differentiating a function that is expressed as a ratio of two differentiable functions. It states that the derivative of a quotient f(t)/g(t) is given by (g(t)f'(t) - f(t)g'(t)) divided by [g(t)]^2. This rule is essential when handling expressions where one function is divided by another.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate a composite function. When a function is defined in terms of other functions, each of which depends on the independent variable, the chain rule allows you to compute the overall derivative by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function.

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