Derivative Rules

Question

Explain the rules applied and find the derivative of the following equation.

A(t) = t²e^5-t

Answer 1

Use the following rules starting with the first one and then using the subsequent rules as applicable.

Product rule of derivatives for (t²)•(e^5-x)

If h(x)=[f(x)]•[g(x)],

then h’(x)=[f(x)]•[g’(x)]+[f’(x)]•[g(x)]

Power rule of derivatives for t²

If f(x)=xⁿ, then f’(x)=nx^n-1

Product [or quotient] rule of exponents (not of derivatives) for e^5-x

bⁿ•b^m=b^n+m

bⁿ/b^m=b^n-m

Product [or quotient] rule of derivatives for e^5-x

• If h(x)=[f(x)]•[g(x)] then

h’(x)=[f(x)]•[g’(x)]+[f’(x)]•[g(x)]

• If h(x)=[f(x)] / [g(x)], then

h’(x)={[g(x)]•[f’(x)]-[f(x)]•[g’(x)]} / {[g(x)]²}

Answer 2

Use the product rule for this problem, i.e., the first times the derivative of the second + the second times the derivative of the first.

problem is: t^2 * e^(5-x)

t^2 is the first, e^(5-x) is the second.

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