How do I take the partial derivative of P with respect to r of the following:
D × P = ∑(t=1 to T)[t × (CF)_t/(1+r)^t ]
To take the partial derivative of P with respect to r in the given expression, you can follow these steps:
1. Start by understanding the notation in the expression:
- D is a constant.
- P and r are variables.
- ∑ represents the summation operator.
- t is the summation index.
- T is the upper limit of the summation.
- CF is a function of t.
2. Expand the expression inside the summation:
- CF_t represents the cash flow at time t.
- (CF)_t/(1+r)^t represents the present value of the cash flow at time t, discounted by the interest rate r.
3. Understand the structure of the equation:
- D × P = ∑(t=1 to T)[t × (CF)_t/(1+r)^t] is an equation relating D, P, r, and the summation term.
4. To find the derivative of P with respect to r, treat P as a function of r and differentiate both sides of the equation with respect to r.
5. Differentiate each term of the equation:
- Differentiating D × P with respect to r gives D × dP/dr (using the product rule).
- Differentiating the summation term gives the derivative of the summation.
6. Apply the chain rule when differentiating the summation term.
- The chain rule states that if you have a composite function like f(g(x)), then the derivative with respect to x is given by f'(g(x)) × g'(x).
- In this case, the composite function is (CF)_t/(1+r)^t, and you need to differentiate it with respect to r.
7. Differentiate (CF)_t with respect to r, keeping t constant.
- Since (CF)_t does not depend on r, its derivative with respect to r is 0.
8. Differentiate (1+r)^t with respect to r, keeping t constant.
- The derivative of (1+r)^t with respect to r is t*(1+r)^(t-1).
9. Substitute the derivatives back into the equation:
- D × dP/dr = ∑(t=1 to T)[t × [(0)/(1+r)^t ] + [(t*(CF)_t)/(1+r)^(t-1)]]
10. Simplify the equation:
- D × dP/dr = ∑(t=1 to T)[ (t*(CF)_t)/(1+r)^(t-1) ]
Therefore, the partial derivative of P with respect to r is given by the equation D × dP/dr = ∑(t=1 to T)[ (t*(CF)_t)/(1+r)^(t-1) ].